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Theorem elfunsALTVfunALTV 37871
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 37593 . . . 4 (𝐹𝑉 → ≀ 𝐹 ∈ V)
2 elcnvrefrelsrel 37710 . . . 4 ( ≀ 𝐹 ∈ V → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
31, 2syl 17 . . 3 (𝐹𝑉 → ( ≀ 𝐹 ∈ CnvRefRels ↔ CnvRefRel ≀ 𝐹))
4 elrelsrel 37661 . . 3 (𝐹𝑉 → (𝐹 ∈ Rels ↔ Rel 𝐹))
53, 4anbi12d 630 . 2 (𝐹𝑉 → (( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ) ↔ ( CnvRefRel ≀ 𝐹 ∧ Rel 𝐹)))
6 elfunsALTV 37866 . 2 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
7 df-funALTV 37856 . 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 395  wcel 2105  Vcvv 3473  Rel wrel 5681  ccoss 37347   Rels crels 37349   CnvRefRels ccnvrefrels 37355   CnvRefRel wcnvrefrel 37356   FunsALTV cfunsALTV 37377   FunALTV wfunALTV 37378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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 37585  df-rels 37659  df-ssr 37672  df-cnvrefs 37699  df-cnvrefrels 37700  df-cnvrefrel 37701  df-funss 37854  df-funsALTV 37855  df-funALTV 37856
This theorem is referenced by: (None)
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