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Theorem funALTVfun 35801
 Description: Our definition of the function predicate df-funALTV 35785 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6353, are always the same and interchangeable. (Contributed by Peter Mazsa, 27-Jul-2021.)
Assertion
Ref Expression
funALTVfun ( FunALTV 𝐹 ↔ Fun 𝐹)

Proof of Theorem funALTVfun
StepHypRef Expression
1 cnvrefrelcoss2 35643 . . . 4 ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I )
2 dfcoss3 35532 . . . . 5 𝐹 = (𝐹𝐹)
32sseq1i 3998 . . . 4 ( ≀ 𝐹 ⊆ I ↔ (𝐹𝐹) ⊆ I )
41, 3bitri 276 . . 3 ( CnvRefRel ≀ 𝐹 ↔ (𝐹𝐹) ⊆ I )
54anbi2ci 624 . 2 (( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹) ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
6 df-funALTV 35785 . 2 ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
7 df-fun 6353 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
85, 6, 73bitr4i 304 1 ( FunALTV 𝐹 ↔ Fun 𝐹)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ wa 396   ⊆ wss 3939   I cid 5457  ◡ccnv 5552   ∘ ccom 5557  Rel wrel 5558  Fun wfun 6345   ≀ ccoss 35324   CnvRefRel wcnvrefrel 35333   FunALTV wfunALTV 35355 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-fun 6353  df-coss 35529  df-cnvrefrel 35635  df-funALTV 35785 This theorem is referenced by: (None)
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