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Theorem elfunsALTVfunALTV 37209
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 36931 . . . 4 (𝐹𝑉 → ≀ 𝐹 ∈ V)
2 elcnvrefrelsrel 37048 . . . 4 ( ≀ 𝐹 ∈ V → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
31, 2syl 17 . . 3 (𝐹𝑉 → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
4 elrelsrel 36999 . . 3 (𝐹𝑉 → (𝐹 ∈ Rels ↔ Rel 𝐹))
53, 4anbi12d 632 . 2 (𝐹𝑉 → (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)))
6 elfunsALTV 37204 . 2 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
7 df-funALTV 37194 . 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 3447  Rel wrel 5642  ccoss 36684   Rels crels 36686   CnvRefRels ccnvrefrels 36692   CnvRefRel wcnvrefrel 36693   FunsALTV cfunsALTV 36714   FunALTV wfunALTV 36715
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-coss 36923  df-rels 36997  df-ssr 37010  df-cnvrefs 37037  df-cnvrefrels 37038  df-cnvrefrel 37039  df-funss 37192  df-funsALTV 37193  df-funALTV 37194
This theorem is referenced by: (None)
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