Theorem elfunsALTV 39350

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.

Step	Hyp	Ref	Expression
1		dffunsALTV 39341	. 2 ⊢ FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels }
2		cosseq 39089	. . 3 ⊢ (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹)
3	2	eleq1d 2854	. 2 ⊢ (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels ))
4	1, 3	rabeqel 38830	1 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ≀ ccoss 38756   Rels crels 38758   CnvRefRels ccnvrefrels 38764   FunsALTV cfunsALTV 38788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-in 3920  df-br 5114  df-opab 5178  df-coss 39074  df-funss 39338  df-funsALTV 39339

This theorem is referenced by:  elfunsALTV2  39351  elfunsALTV3  39352  elfunsALTV4  39353  elfunsALTV5  39354  elfunsALTVfunALTV  39355