Theorem frege122 43943

Description: If 𝑋 is a result of an application of the single-valued procedure 𝑅 to 𝑌, then every result of an application of the procedure 𝑅 to 𝑌 belongs to the 𝑅-sequence beginning with 𝑋. Proposition 122 of [Frege1879] p. 79. (Contributed by RP, 8-Jul-2020.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
frege116.x	⊢ 𝑋 ∈ 𝑈
frege118.y	⊢ 𝑌 ∈ 𝑉
frege120.a	⊢ 𝐴 ∈ 𝑊

Assertion

Ref	Expression
frege122	⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))

Proof of Theorem frege122

Step	Hyp	Ref	Expression
1		frege120.a	. . 3 ⊢ 𝐴 ∈ 𝑊
2	1	frege112 43933	. 2 ⊢ (𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴)
3		frege116.x	. . 3 ⊢ 𝑋 ∈ 𝑈
4		frege118.y	. . 3 ⊢ 𝑌 ∈ 𝑉
5	3, 4, 1	frege121 43942	. 2 ⊢ ((𝐴 = 𝑋 → 𝑋((t+‘𝑅) ∪ I )𝐴) → (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴))))
6	2, 5	ax-mp 5	1 ⊢ (Fun ◡◡𝑅 → (𝑌𝑅𝑋 → (𝑌𝑅𝐴 → 𝑋((t+‘𝑅) ∪ I )𝐴)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ∪ cun 3931   class class class wbr 5125   I cid 5559  ^◡ccnv 5666  Fun wfun 6536  ‘cfv 6542  t+ctcl 15007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pr 5414  ax-frege1 43748  ax-frege2 43749  ax-frege8 43767  ax-frege52a 43815  ax-frege58b 43859

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-ss 3950  df-nul 4316  df-if 4508  df-sn 4609  df-pr 4611  df-op 4615  df-br 5126  df-opab 5188  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-fun 6544

This theorem is referenced by:  frege123  43944