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Theorem elfunsALTVfunALTV 37567
Description: The element of the class of functions and the function predicate are the same when 𝐹 is a set. (Contributed by Peter Mazsa, 26-Jul-2021.)
Assertion
Ref Expression
elfunsALTVfunALTV (𝐹𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹))

Proof of Theorem elfunsALTVfunALTV
StepHypRef Expression
1 cossex 37289 . . . 4 (𝐹𝑉 → ≀ 𝐹 ∈ V)
2 elcnvrefrelsrel 37406 . . . 4 ( ≀ 𝐹 ∈ V → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
31, 2syl 17 . . 3 (𝐹𝑉 → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
4 elrelsrel 37357 . . 3 (𝐹𝑉 → (𝐹 ∈ Rels ↔ Rel 𝐹))
53, 4anbi12d 632 . 2 (𝐹𝑉 → (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)))
6 elfunsALTV 37562 . 2 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
7 df-funALTV 37552 . 2 ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
85, 6, 73bitr4g 314 1 (𝐹𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  Vcvv 3475  Rel wrel 5682  ccoss 37043   Rels crels 37045   CnvRefRels ccnvrefrels 37051   CnvRefRel wcnvrefrel 37052   FunsALTV cfunsALTV 37073   FunALTV wfunALTV 37074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-coss 37281  df-rels 37355  df-ssr 37368  df-cnvrefs 37395  df-cnvrefrels 37396  df-cnvrefrel 37397  df-funss 37550  df-funsALTV 37551  df-funALTV 37552
This theorem is referenced by: (None)
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