Theorem elfunsALTV 38674

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 38665	. 2 ⊢ FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels }
2		cosseq 38408	. . 3 ⊢ (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹)
3	2	eleq1d 2824	. 2 ⊢ (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels ))
4	1, 3	rabeqel 38236	1 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels ))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≀ ccoss 38162   Rels crels 38164   CnvRefRels ccnvrefrels 38170   FunsALTV cfunsALTV 38192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970  df-br 5149  df-opab 5211  df-coss 38393  df-funss 38662  df-funsALTV 38663

This theorem is referenced by:  elfunsALTV2  38675  elfunsALTV3  38676  elfunsALTV4  38677  elfunsALTV5  38678  elfunsALTVfunALTV  38679