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Theorem elfunsALTVfunALTV 37436
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 37158 . . . 4 (𝐹𝑉 → ≀ 𝐹 ∈ V)
2 elcnvrefrelsrel 37275 . . . 4 ( ≀ 𝐹 ∈ V → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
31, 2syl 17 . . 3 (𝐹𝑉 → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
4 elrelsrel 37226 . . 3 (𝐹𝑉 → (𝐹 ∈ Rels ↔ Rel 𝐹))
53, 4anbi12d 631 . 2 (𝐹𝑉 → (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)))
6 elfunsALTV 37431 . 2 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
7 df-funALTV 37421 . 2 ( FunALTV 𝐹 ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹))
85, 6, 73bitr4g 313 1 (𝐹𝑉 → (𝐹 ∈ FunsALTV ↔ FunALTV 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  Vcvv 3474  Rel wrel 5675  ccoss 36912   Rels crels 36914   CnvRefRels ccnvrefrels 36920   CnvRefRel wcnvrefrel 36921   FunsALTV cfunsALTV 36942   FunALTV wfunALTV 36943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-coss 37150  df-rels 37224  df-ssr 37237  df-cnvrefs 37264  df-cnvrefrels 37265  df-cnvrefrel 37266  df-funss 37419  df-funsALTV 37420  df-funALTV 37421
This theorem is referenced by: (None)
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