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Theorem elfunsALTV 36829
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 36820 . 2 FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels }
2 cosseq 36575 . . 3 (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹)
32eleq1d 2818 . 2 (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels ))
41, 3rabeqel 36421 1 (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1537  wcel 2101  ccoss 36361   Rels crels 36363   CnvRefRels ccnvrefrels 36369   FunsALTV cfunsALTV 36391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1540  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3224  df-v 3436  df-in 3896  df-br 5078  df-opab 5140  df-coss 36563  df-funss 36817  df-funsALTV 36818
This theorem is referenced by:  elfunsALTV2  36830  elfunsALTV3  36831  elfunsALTV4  36832  elfunsALTV5  36833  elfunsALTVfunALTV  36834
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