Theorem elfunsALTV 39116

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 39107	. 2 ⊢ FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels }
2		cosseq 38855	. . 3 ⊢ (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹)
3	2	eleq1d 2822	. 2 ⊢ (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels ))
4	1, 3	rabeqel 38596	1 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≀ ccoss 38522   Rels crels 38524   CnvRefRels ccnvrefrels 38530   FunsALTV cfunsALTV 38554

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-in 3897  df-br 5087  df-opab 5149  df-coss 38840  df-funss 39104  df-funsALTV 39105

This theorem is referenced by:  elfunsALTV2  39117  elfunsALTV3  39118  elfunsALTV4  39119  elfunsALTV5  39120  elfunsALTVfunALTV  39121