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Theorem dffunALTV3 36362
 Description: Alternate definition of the function relation predicate, cf. dfdisjALTV3 36388. Reproduction of dffun2 6345. For the 𝑋 axis and the 𝑌 axis you can convert the right side to (∀ x1 ∀ y1 ∀ y2 (( x1 𝑓 y1 ∧ x1 𝑓 y2 ) → y1 = y2 ) ∧ Rel 𝐹). (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dffunALTV3 ( FunALTV 𝐹 ↔ (∀𝑢𝑥𝑦((𝑢𝐹𝑥𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))
Distinct variable group:   𝑢,𝐹,𝑥,𝑦

Proof of Theorem dffunALTV3
StepHypRef Expression
1 dffunALTV2 36361 . 2 ( FunALTV 𝐹 ↔ ( ≀ 𝐹 ⊆ I ∧ Rel 𝐹))
2 cossssid3 36149 . . 3 ( ≀ 𝐹 ⊆ I ↔ ∀𝑢𝑥𝑦((𝑢𝐹𝑥𝑢𝐹𝑦) → 𝑥 = 𝑦))
32anbi1i 626 . 2 (( ≀ 𝐹 ⊆ I ∧ Rel 𝐹) ↔ (∀𝑢𝑥𝑦((𝑢𝐹𝑥𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))
41, 3bitri 278 1 ( FunALTV 𝐹 ↔ (∀𝑢𝑥𝑦((𝑢𝐹𝑥𝑢𝐹𝑦) → 𝑥 = 𝑦) ∧ Rel 𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   ⊆ wss 3858   class class class wbr 5032   I cid 5429  Rel wrel 5529   ≀ ccoss 35893   FunALTV wfunALTV 35924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-coss 36099  df-cnvrefrel 36205  df-funALTV 36355 This theorem is referenced by: (None)
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