Theorem elfunsALTV 38890

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 38881	. 2 ⊢ FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels }
2		cosseq 38628	. . 3 ⊢ (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹)
3	2	eleq1d 2819	. 2 ⊢ (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels ))
4	1, 3	rabeqel 38391	1 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≀ ccoss 38322   Rels crels 38324   CnvRefRels ccnvrefrels 38330   FunsALTV cfunsALTV 38352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-in 3906  df-br 5097  df-opab 5159  df-coss 38613  df-funss 38878  df-funsALTV 38879

This theorem is referenced by:  elfunsALTV2  38891  elfunsALTV3  38892  elfunsALTV4  38893  elfunsALTV5  38894  elfunsALTVfunALTV  38895