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Theorem elfunsALTVfunALTV 37562
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 37284 . . . 4 (𝐹𝑉 → ≀ 𝐹 ∈ V)
2 elcnvrefrelsrel 37401 . . . 4 ( ≀ 𝐹 ∈ V → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
31, 2syl 17 . . 3 (𝐹𝑉 → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
4 elrelsrel 37352 . . 3 (𝐹𝑉 → (𝐹 ∈ Rels ↔ Rel 𝐹))
53, 4anbi12d 631 . 2 (𝐹𝑉 → (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)))
6 elfunsALTV 37557 . 2 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
7 df-funALTV 37547 . 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 5681  ccoss 37038   Rels crels 37040   CnvRefRels ccnvrefrels 37046   CnvRefRel wcnvrefrel 37047   FunsALTV cfunsALTV 37068   FunALTV wfunALTV 37069
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-coss 37276  df-rels 37350  df-ssr 37363  df-cnvrefs 37390  df-cnvrefrels 37391  df-cnvrefrel 37392  df-funss 37545  df-funsALTV 37546  df-funALTV 37547
This theorem is referenced by: (None)
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