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Theorem List for Metamath Proof Explorer - 40901-41000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmdandyvr6 40901 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvr7 40902 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvr8 40903 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvr9 40904 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvr10 40905 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvr11 40906 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvr12 40907 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvr13 40908 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvr14 40909 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvr15 40910 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx0 40911 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvrx1 40912 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvrx2 40913 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvrx3 40914 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜁))

Theoremmdandyvrx4 40915 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx5 40916 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx6 40917 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx7 40918 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜑)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜁))

Theoremmdandyvrx8 40919 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx9 40920 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx10 40921 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx11 40922 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜑)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜁)) ∧ (𝜂𝜎))

Theoremmdandyvrx12 40923 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx13 40924 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜑)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜁)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx14 40925 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜑)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜁) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

Theoremmdandyvrx15 40926 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016.)
(𝜑𝜁)    &   (𝜓𝜎)    &   (𝜒𝜓)    &   (𝜃𝜓)    &   (𝜏𝜓)    &   (𝜂𝜓)       ((((𝜒𝜎) ∧ (𝜃𝜎)) ∧ (𝜏𝜎)) ∧ (𝜂𝜎))

TheoremH15NH16TH15IH16 40927 Given 15 hypotheses and a 16th hypothesis, there exists a proof the 15 imply the 16th. (Contributed by Jarvin Udandy, 8-Sep-2016.)
𝜑    &   𝜓    &   𝜒    &   𝜃    &   𝜏    &   𝜂    &   𝜁    &   𝜎    &   𝜌    &   𝜇    &   𝜆    &   𝜅    &   jph    &   jps    &   jch    &   jth       (((((((((((((((𝜑𝜓) ∧ 𝜒) ∧ 𝜃) ∧ 𝜏) ∧ 𝜂) ∧ 𝜁) ∧ 𝜎) ∧ 𝜌) ∧ 𝜇) ∧ 𝜆) ∧ 𝜅) ∧ jph) ∧ jps) ∧ jch) → jth)

Theoremdandysum2p2e4 40928

CONTRADICTION PROVED AT 1 + 1 = 2 .

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit. (Contributed by Jarvin Udandy, 6-Sep-2016.)

(𝜑 ↔ (𝜃𝜏))    &   (𝜓 ↔ (𝜂𝜁))    &   (𝜒 ↔ (𝜎𝜌))    &   (𝜃 ↔ ⊥)    &   (𝜏 ↔ ⊥)    &   (𝜂 ↔ ⊤)    &   (𝜁 ↔ ⊤)    &   (𝜎 ↔ ⊥)    &   (𝜌 ↔ ⊥)    &   (𝜇 ↔ ⊥)    &   (𝜆 ↔ ⊥)    &   (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))    &   (jph ↔ ((𝜂𝜁) ∨ 𝜑))    &   (jps ↔ ((𝜎𝜌) ∨ 𝜓))    &   (jch ↔ ((𝜇𝜆) ∨ 𝜒))       ((((((((((((((((𝜑 ↔ (𝜃𝜏)) ∧ (𝜓 ↔ (𝜂𝜁))) ∧ (𝜒 ↔ (𝜎𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))) ∧ (jph ↔ ((𝜂𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥)))

Theoremmdandysum2p2e4 40929 CONTRADICTION PROVED AT 1 + 1 = 2 . Luckily Mario Carneiro did a successful version of his own.

See Mario's Relevant Work: 1.3.14 Half adder and full adder in propositional calculus.

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F. ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit.

In mdandysum2p2e4, one might imagine what jth or jta could be then do the math with their truths. Also limited to the restriction jth, jta are having opposite truths equivalent to the stated truth constants.

(Contributed by Jarvin Udandy, 6-Sep-2016.)

(jth ↔ ⊥)    &   (jta ↔ ⊤)    &   (𝜑 ↔ (𝜃𝜏))    &   (𝜓 ↔ (𝜂𝜁))    &   (𝜒 ↔ (𝜎𝜌))    &   (𝜃jth)    &   (𝜏jth)    &   (𝜂jta)    &   (𝜁jta)    &   (𝜎jth)    &   (𝜌jth)    &   (𝜇jth)    &   (𝜆jth)    &   (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))    &   (jph ↔ ((𝜂𝜁) ∨ 𝜑))    &   (jps ↔ ((𝜎𝜌) ∨ 𝜓))    &   (jch ↔ ((𝜇𝜆) ∨ 𝜒))       ((((((((((((((((𝜑 ↔ (𝜃𝜏)) ∧ (𝜓 ↔ (𝜂𝜁))) ∧ (𝜒 ↔ (𝜎𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃𝜏) ⊻ (𝜃𝜏)))) ∧ (jph ↔ ((𝜂𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥)))

20.34  Mathbox for Alexander van der Vekens

20.34.1  Double restricted existential uniqueness

20.34.1.1  Restricted quantification (extension)

Theoremr19.32 40930 Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 3078. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
𝑥𝜑       (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝐴 𝜓))

Theoremrexsb 40931* An equivalent expression for restricted existence, analogous to exsb 2466. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥(𝑥 = 𝑦𝜑))

Theoremrexrsb 40932* An equivalent expression for restricted existence, analogous to exsb 2466. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝑥 = 𝑦𝜑))

Theorem2rexsb 40933* An equivalent expression for double restricted existence, analogous to rexsb 40931. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theorem2rexrsb 40934* An equivalent expression for double restricted existence, analogous to 2exsb 2467. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
(∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))

Theoremcbvral2 40935* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 3174. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑧𝜑    &   𝑥𝜒    &   𝑤𝜒    &   𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 ↔ ∀𝑧𝐴𝑤𝐵 𝜓)

Theoremcbvrex2 40936* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3175. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑧𝜑    &   𝑥𝜒    &   𝑤𝜒    &   𝑦𝜓    &   (𝑥 = 𝑧 → (𝜑𝜒))    &   (𝑦 = 𝑤 → (𝜒𝜓))       (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)

Theorem2ralbiim 40937 Split a biconditional and distribute 2 quantifiers, analogous to 2albiim 1815 and ralbiim 3065. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∀𝑥𝐴𝑦𝐵 (𝜓𝜑)))

20.34.1.2  The empty set (extension)

Theoremraaan2 40938* Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 4073. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
𝑦𝜑    &   𝑥𝜓       ((𝐴 = ∅ ↔ 𝐵 = ∅) → (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐵 𝜓)))

20.34.1.3  Restricted uniqueness and "at most one" quantification

Theoremrmoimi 40939 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(𝜑𝜓)       (∃*𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑)

Theorem2reu5a 40940 Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑) ∧ ∃*𝑥𝐴 (∃𝑦𝐵 𝜑 ∧ ∃*𝑦𝐵 𝜑)))

Theoremreuimrmo 40941 Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2520. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 (𝜑𝜓) → (∃!𝑥𝐴 𝜓 → ∃*𝑥𝐴 𝜑))

Theoremrmoanim 40942* Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2527. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
𝑥𝜑       (∃*𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃*𝑥𝐴 𝜓))

Theoremreuan 40943* Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2528. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
𝑥𝜑       (∃!𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∃!𝑥𝐴 𝜓))

20.34.1.4  Analogs to Existential uniqueness (double quantification)

Theorem2reurex 40944* Double restricted quantification with existential uniqueness, analogous to 2euex 2542. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
(∃!𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦𝐵 ∃!𝑥𝐴 𝜑)

Theorem2reurmo 40945* Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2543. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
(∃!𝑥𝐴 ∃*𝑦𝐵 𝜑 → ∃*𝑥𝐴 ∃!𝑦𝐵 𝜑)

Theorem2reu2rex 40946* Double restricted existential uniqueness, analogous to 2eu2ex 2544. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 → ∃𝑥𝐴𝑦𝐵 𝜑)

Theorem2rmoswap 40947* A condition allowing swap of restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2545. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃*𝑥𝐴𝑦𝐵 𝜑 → ∃*𝑦𝐵𝑥𝐴 𝜑))

Theorem2rexreu 40948* Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2547. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) → ∃!𝑥𝐴 ∃!𝑦𝐵 𝜑)

Theorem2reu1 40949* Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2551. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
(∀𝑥𝐴 ∃*𝑦𝐵 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))

Theorem2reu2 40950* Double restricted existential uniqueness, analogous to 2eu2 2552. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
(∃!𝑦𝐵𝑥𝐴 𝜑 → (∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ↔ ∃!𝑥𝐴𝑦𝐵 𝜑))

Theorem2reu3 40951* Double restricted existential uniqueness, analogous to 2eu3 2553. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
(∀𝑥𝐴𝑦𝐵 (∃*𝑥𝐴 𝜑 ∨ ∃*𝑦𝐵 𝜑) → ((∃!𝑥𝐴 ∃!𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵 ∃!𝑥𝐴 𝜑) ↔ (∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑)))

Theorem2reu4a 40952* Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2554 with the additional requirement that the restricting classes are not empty (which is not necessary as shown in 2reu4 40953). (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤)))))

Theorem2reu4 40953* Definition of double restricted existential uniqueness ("exactly one 𝑥 and exactly one 𝑦"), analogous to 2eu4 2554. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ (∃𝑥𝐴𝑦𝐵 𝜑 ∧ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 (𝜑 → (𝑥 = 𝑧𝑦 = 𝑤))))

Theorem2reu7 40954* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2557. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
((∃!𝑥𝐴𝑦𝐵 𝜑 ∧ ∃!𝑦𝐵𝑥𝐴 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))

Theorem2reu8 40955* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2558. Curiously, we can put ∃! on either of the internal conjuncts but not both. We can also commute ∃!𝑥𝐴∃!𝑦𝐵 using 2reu7 40954. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(∃!𝑥𝐴 ∃!𝑦𝐵 (∃𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑) ↔ ∃!𝑥𝐴 ∃!𝑦𝐵 (∃!𝑥𝐴 𝜑 ∧ ∃𝑦𝐵 𝜑))

20.34.2  Alternative definitions of function and operation values

The current definition of the value (𝐹𝐴) of a function 𝐹 at an argument 𝐴 (see df-fv 5884) assures that this value is always a set, see fex 6475. This is because this definition can be applied to any classes 𝐹 and 𝐴, and evaluates to the empty set when it is not meaningful (as shown by ndmfv 6205 and fvprc 6172).

Although it is very convenient for many theorems on functions and their proofs, there are some cases in which from (𝐹𝐴) = ∅ alone it cannot be decided/derived whether (𝐹𝐴) is meaningful (𝐹 is actually a function which is defined for 𝐴 and really has the function value at 𝐴) or not. Therefore, additional assumptions are required, such as ∅ ∉ ran 𝐹, ∅ ∈ ran 𝐹 or Fun 𝐹𝐴 ∈ dom 𝐹 (see, for example, ndmfvrcl 6206).

To avoid such an ambiguity, an alternative definition (𝐹'''𝐴) (see df-afv 40960) would be possible which evaluates to the universal class ((𝐹'''𝐴) = V) if it is not meaningful (see afvnfundmuv 40982, ndmafv 40983, afvprc 40987 and nfunsnafv 40985), and which corresponds to the current definition ((𝐹𝐴) = (𝐹'''𝐴)) if it is (see afvfundmfveq 40981). That means (𝐹'''𝐴) = V → (𝐹𝐴) = ∅ (see afvpcfv0 40989), but (𝐹𝐴) = ∅ → (𝐹'''𝐴) = V is not generally valid.

In the theory of partial functions, it is a common case that 𝐹 is not defined at 𝐴, which also would result in (𝐹'''𝐴) = V. In this context we say (𝐹'''𝐴) "is not defined" instead of "is not meaningful".

With this definition the following intuitive equivalence holds: (𝐹'''𝐴) ∈ V <-> "(𝐹'''𝐴) is meaningful/defined".

An interesting question would be if (𝐹𝐴) could be replaced by (𝐹'''𝐴) in most of the theorems based on function values. If we look at the (currently 19) proofs using the definition df-fv 5884 of (𝐹𝐴), we see that analogues for the following 8 theorems can be proven using the alternative definition: fveq1 6177-> afveq1 40977, fveq2 6178-> afveq2 40978, nffv 6185-> nfafv 40979, csbfv12 6218-> csbafv12g , fvres 6194-> afvres 41015, rlimdm 14263-> rlimdmafv 41020, tz6.12-1 6197-> tz6.12-1-afv 41017, fveu 6170-> afveu 40996.

Three theorems proved by directly using df-fv 5884 are within a mathbox (fvsb 38476) or not used (isumclim3 14471, avril1 27289).

However, the remaining 8 theorems proved by directly using df-fv 5884 are used more or less often:

* fvex 6188: used in about 1750 proofs.

* tz6.12-1 6197: root theorem of many theorems which have not a strict analogue, and which are used many times: fvprc 6172 (used in about 127 proofs), tz6.12i 6201 (used - indirectly via fvbr0 6202 and fvrn0 6203- in 18 proofs, and in fvclss 6485 used in fvclex 7123 used in fvresex 7124, which is not used!), dcomex 9254 (used in 4 proofs), ndmfv 6205 (used in 86 proofs) and nfunsn 6212 (used by dffv2 6258 which is not used).

* fv2 6173: only used by elfv 6176, which is only used by fv3 6193, which is not used.

* dffv3 6174: used by dffv4 6175 (the previous "df-fv"), which now is only used in deprecated (usage discouraged) theorems or within mathboxes (csbfv12gALTOLD 38872, csbfv12gALTVD 38955), by shftval 13795 (itself used in 9 proofs), by dffv5 32006 (mathbox) and by fvco2 6260, which has the analogue afvco2 41019.

* fvopab5 6295: used only by ajval 27687 (not used) and by adjval 28719 ( used - indirectly - in 9 proofs).

* zsum 14430: used (via isum 14431, sum0 14433 and fsumsers 14440) in more than 90 proofs.

* isumshft 14552: used in pserdv2 24165 and (via logtayl 24387) 4 other proofs.

* ovtpos 7352: used in 14 proofs.

As a result of this analysis we can say that the current definition of a function value is crucial for Metamath and cannot be exchanged easily with an alternative definition. While fv2 6173, dffv3 6174, fvopab5 6295, zsum 14430, isumshft 14552 and ovtpos 7352 are not critical or are, hopefully, also valid for the alternative definition, fvex 6188 and tz6.12-1 6197 (and the theorems based on them) are essential for the current definition of function values.

With the same arguments, an alternative definition of operation values ((𝐴𝑂𝐵)) could be meaningful to avoid ambiguities, see df-aov 40961.

Syntaxwdfat 40956 Extend the definition of a wff to include the "defined at" predicate. (Read: (The Function) 𝐹 is defined at (the argument) 𝐴). In a previous version, the token "def@" was used. However, since the @ is used (informally) as a replacement for \$ in commented out sections that may be deleted some day. While there is no violation of any standard to use the @ in a token, it could make the search for such commented-out sections slightly more difficult. (See remark of Norman Megill at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
wff 𝐹 defAt 𝐴

Syntaxcafv 40957 Extend the definition of a class to include the value of a function. (Read: The value of 𝐹 at 𝐴, or "𝐹 of 𝐴."). In a previous version, the symbol " ' " was used. However, since the similarity with the symbol used for the current definition of a function's value (see df-fv 5884), which, by the way, was intended to visualize that in many cases and " ' " are exchangeable, makes reading the theorems, especially those which uses both definitions as dfafv2 40975, very difficult, 3 apostrophes ''' are used now so that it's easier to distinguish from df-fv 5884 and df-ima 5117. And not three backticks ( three times ) since that would be annoying to escape in a comment. (See remark of Norman Megill and Gerard Lang at https://groups.google.com/forum/#!topic/metamath/cteNUppB6A4).
class (𝐹'''𝐴)

Syntaxcaov 40958 Extend class notation to include the value of an operation 𝐹 (such as +) for two arguments 𝐴 and 𝐵. Note that the syntax is simply three class symbols in a row surrounded by a pair of parentheses in contrast to the current definition, see df-ov 6638.
class ((𝐴𝐹𝐵))

Definitiondf-dfat 40959 Definition of the predicate that determines if some class 𝐹 is defined as function for an argument 𝐴 or, in other words, if the function value for some class 𝐹 for an argument 𝐴 is defined. We say that 𝐹 is defined at 𝐴 if a 𝐹 is a function restricted to the member 𝐴 of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Definitiondf-afv 40960* Alternative definition of the value of a function, (𝐹'''𝐴), also known as function application. In contrast to (𝐹𝐴) = ∅ (see df-fv 5884 and ndmfv 6205), (𝐹'''𝐴) = V if F is not defined for A! (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹'''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), V)

Definitiondf-aov 40961 Define the value of an operation. In contrast to df-ov 6638, the alternative definition for a function value (see df-afv 40960) is used. By this, the value of the operation applied to two arguments is the universal class if the operation is not defined for these two arguments. There are still no restrictions of any kind on what those class expressions may be, although only certain kinds of class expressions - a binary operation 𝐹 and its arguments 𝐴 and 𝐵- will be useful for proving meaningful theorems. (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐴𝐹𝐵)) = (𝐹'''⟨𝐴, 𝐵⟩)

20.34.2.1  Restricted quantification (extension)

Theoremralbinrald 40962* Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
(𝜑𝑋𝐴)    &   (𝑥𝐴𝑥 = 𝑋)    &   (𝑥 = 𝑋 → (𝜓𝜃))       (𝜑 → (∀𝑥𝐴 𝜓𝜃))

20.34.2.2  The universal class (extension)

Theoremnvelim 40963 If a class is the universal class it doesn't belong to any class, generalisation of nvel 4788. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝐴 = V → ¬ 𝐴𝐵)

20.34.2.3  Introduce the Axiom of Power Sets (extension)

Theoremalneu 40964 If a statement holds for all sets, there is not a unique set for which the statement holds. (Contributed by Alexander van der Vekens, 28-Nov-2017.)
(∀𝑥𝜑 → ¬ ∃!𝑥𝜑)

Theoremeu2ndop1stv 40965* If there is a unique second component in an ordered pair contained in a given set, the first component must be a set. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑦𝐴, 𝑦⟩ ∈ 𝑉𝐴 ∈ V)

20.34.2.4  Relations (extension)

Theoremeldmressn 40966 Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
(𝐵 ∈ dom (𝐹 ↾ {𝐴}) → 𝐵 = 𝐴)

20.34.2.5  Functions (extension)

Theoremfveqvfvv 40967 If a function's value at an argument is the universal class (which can never be the case because of fvex 6188), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything (see pm2.21i 116). (Contributed by Alexander van der Vekens, 26-May-2017.)
((𝐹𝐴) = V → (𝐹𝐴) = 𝐵)

Theoremfunresfunco 40968 Composition of two functions, generalization of funco 5916. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹𝐺))

Theoremfnresfnco 40969 Composition of two functions, similar to fnco 5987. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
(((𝐹 ↾ ran 𝐺) Fn ran 𝐺𝐺 Fn 𝐵) → (𝐹𝐺) Fn 𝐵)

Theoremfuncoressn 40970 A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))

Theoremfunressnfv 40971 A restriction to a singleton with a function value is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
(((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ↾ {(𝐺𝑋)}))

20.34.2.6  Predicate "defined at"

Theoremdfateq12d 40972 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))

Theoremnfdfat 40973 Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g., for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥 𝐹 defAt 𝐴

Theoremdfdfat2 40974* Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑦 𝐴𝐹𝑦))

20.34.2.7  Alternative definition of the value of a function

Theoremdfafv2 40975 Alternative definition of (𝐹'''𝐴) using (𝐹𝐴) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)

Theoremafveq12d 40976 Equality deduction for function value, analogous to fveq12d 6184. (Contributed by Alexander van der Vekens, 26-May-2017.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵))

Theoremafveq1 40977 Equality theorem for function value, analogous to fveq1 6177. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐹 = 𝐺 → (𝐹'''𝐴) = (𝐺'''𝐴))

Theoremafveq2 40978 Equality theorem for function value, analogous to fveq1 6177. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
(𝐴 = 𝐵 → (𝐹'''𝐴) = (𝐹'''𝐵))

Theoremnfafv 40979 Bound-variable hypothesis builder for function value, analogous to nffv 6185. To prove a deduction version of this analogous to nffvd 6187 is not easily possible because a deduction version of nfdfat 40973 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹'''𝐴)

Theoremcsbafv12g 40980 Move class substitution in and out of a function value, analogous to csbfv12 6218, with a direct proof proposed by Mario Carneiro, analogous to csbov123 6672. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))

Theoremafvfundmfveq 40981 If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.)
(𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafvnfundmuv 40982 If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)
𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)

Theoremndmafv 40983 The value of a class outside its domain is the universe, compare with ndmfv 6205. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)

Theoremafvvdm 40984 If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ dom 𝐹)

Theoremnfunsnafv 40985 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 6212. (Contributed by Alexander van der Vekens, 25-May-2017.)
(¬ Fun (𝐹 ↾ {𝐴}) → (𝐹'''𝐴) = V)

Theoremafvvfunressn 40986 If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵 → Fun (𝐹 ↾ {𝐴}))

Theoremafvprc 40987 A function's value at a proper class is the universe, compare with fvprc 6172. (Contributed by Alexander van der Vekens, 25-May-2017.)
𝐴 ∈ V → (𝐹'''𝐴) = V)

Theoremafvvv 40988 If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵𝐴 ∈ V)

Theoremafvpcfv0 40989 If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)

Theoremafvnufveq 40990 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafvvfveq 40991 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) ∈ 𝐵 → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafv0fv0 40992 If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)

Theoremafvfvn0fveq 40993 If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹𝐴) ≠ ∅ → (𝐹'''𝐴) = (𝐹𝐴))

Theoremafv0nbfvbi 40994 The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
(∅ ∉ 𝐵 → ((𝐹'''𝐴) ∈ 𝐵 ↔ (𝐹𝐴) ∈ 𝐵))

Theoremafvfv0bi 40995 The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))

Theoremafveu 40996* The value of a function at a unique point, analogous to fveu 6170. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})

Theoremfnbrafvb 40997 Equivalence of function value and binary relation, analogous to fnbrfvb 6223. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))

Theoremfnopafvb 40998 Equivalence of function value and ordered pair membership, analogous to fnopfvb 6224. (Contributed by Alexander van der Vekens, 25-May-2017.)
((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))

Theoremfunbrafvb 40999 Equivalence of function value and binary relation, analogous to funbrfvb 6225. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfunopafvb 41000 Equivalence of function value and ordered pair membership, analogous to funopfvb 6226. (Contributed by Alexander van der Vekens, 25-May-2017.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹'''𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))

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