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Theorem elfunsALTV 36791
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 36782 . 2 FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels }
2 cosseq 36537 . . 3 (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹)
32eleq1d 2825 . 2 (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels ))
41, 3rabeqel 36382 1 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1542  wcel 2110  ccoss 36321   Rels crels 36323   CnvRefRels ccnvrefrels 36329   FunsALTV cfunsALTV 36351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-in 3899  df-br 5080  df-opab 5142  df-coss 36525  df-funss 36779  df-funsALTV 36780
This theorem is referenced by:  elfunsALTV2  36792  elfunsALTV3  36793  elfunsALTV4  36794  elfunsALTV5  36795  elfunsALTVfunALTV  36796
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