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Theorem funALTVfun 38699
Description: Our definition of the function predicate df-funALTV 38683 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6563, 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 38538 . . . 4 ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I )
2 dfcoss3 38415 . . . . 5 𝐹 = (𝐹𝐹)
32sseq1i 4012 . . . 4 ( ≀ 𝐹 ⊆ I ↔ (𝐹𝐹) ⊆ I )
41, 3bitri 275 . . 3 ( CnvRefRel ≀ 𝐹 ↔ (𝐹𝐹) ⊆ I )
54anbi2ci 625 . 2 (( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹) ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
6 df-funALTV 38683 . 2 ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
7 df-fun 6563 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹𝐹) ⊆ I ))
85, 6, 73bitr4i 303 1 ( FunALTV 𝐹 ↔ Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wss 3951   I cid 5577  ccnv 5684  ccom 5689  Rel wrel 5690  Fun wfun 6555  ccoss 38182   CnvRefRel wcnvrefrel 38191   FunALTV wfunALTV 38213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-fun 6563  df-coss 38412  df-cnvrefrel 38528  df-funALTV 38683
This theorem is referenced by: (None)
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