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Theorem elfunsALTV 38693
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 38684 . 2 FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels }
2 cosseq 38427 . . 3 (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹)
32eleq1d 2826 . 2 (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels ))
41, 3rabeqel 38255 1 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  ccoss 38182   Rels crels 38184   CnvRefRels ccnvrefrels 38190   FunsALTV cfunsALTV 38212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-br 5144  df-opab 5206  df-coss 38412  df-funss 38681  df-funsALTV 38682
This theorem is referenced by:  elfunsALTV2  38694  elfunsALTV3  38695  elfunsALTV4  38696  elfunsALTV5  38697  elfunsALTVfunALTV  38698
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