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Theorem funALTVfun 38680
Description: Our definition of the function predicate df-funALTV 38664 (based on a more general, converse reflexive, relation) and the original definition of function in set.mm df-fun 6565, 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 38519 . . . 4 ( CnvRefRel ≀ 𝐹 ↔ ≀ 𝐹 ⊆ I )
2 dfcoss3 38396 . . . . 5 𝐹 = (𝐹𝐹)
32sseq1i 4024 . . . 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 38664 . 2 ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
7 df-fun 6565 . 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 3963   I cid 5582  ccnv 5688  ccom 5693  Rel wrel 5694  Fun wfun 6557  ccoss 38162   CnvRefRel wcnvrefrel 38171   FunALTV wfunALTV 38193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-fun 6565  df-coss 38393  df-cnvrefrel 38509  df-funALTV 38664
This theorem is referenced by: (None)
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