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Theorem elfunsALTV 37654
Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.)
Assertion
Ref Expression
elfunsALTV (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))

Proof of Theorem elfunsALTV
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dffunsALTV 37645 . 2 FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels }
2 cosseq 37388 . . 3 (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹)
32eleq1d 2818 . 2 (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels ))
41, 3rabeqel 37214 1 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  ccoss 37135   Rels crels 37137   CnvRefRels ccnvrefrels 37143   FunsALTV cfunsALTV 37165
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-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-in 3955  df-br 5149  df-opab 5211  df-coss 37373  df-funss 37642  df-funsALTV 37643
This theorem is referenced by:  elfunsALTV2  37655  elfunsALTV3  37656  elfunsALTV4  37657  elfunsALTV5  37658  elfunsALTVfunALTV  37659
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