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.

Step	Hyp	Ref	Expression
1		dffunsALTV 36820	. 2 ⊢ FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels }
2		cosseq 36575	. . 3 ⊢ (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹)
3	2	eleq1d 2818	. 2 ⊢ (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels ))
4	1, 3	rabeqel 36421	1 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))

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