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Mirrors > Home > MPE Home > Th. List > Mathboxes > elfunsALTV | Structured version Visualization version GIF version |
Description: Elementhood in the class of functions. (Contributed by Peter Mazsa, 24-Jul-2021.) |
Ref | Expression |
---|---|
elfunsALTV | ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffunsALTV 37645 | . 2 ⊢ FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels } | |
2 | cosseq 37388 | . . 3 ⊢ (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹) | |
3 | 2 | eleq1d 2818 | . 2 ⊢ (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels )) |
4 | 1, 3 | rabeqel 37214 | 1 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≀ ccoss 37135 Rels crels 37137 CnvRefRels ccnvrefrels 37143 FunsALTV cfunsALTV 37165 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-rab 3433 df-v 3476 df-in 3955 df-br 5149 df-opab 5211 df-coss 37373 df-funss 37642 df-funsALTV 37643 |
This theorem is referenced by: elfunsALTV2 37655 elfunsALTV3 37656 elfunsALTV4 37657 elfunsALTV5 37658 elfunsALTVfunALTV 37659 |
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