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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 38684 | . 2 ⊢ FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels } | |
| 2 | cosseq 38427 | . . 3 ⊢ (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹) | |
| 3 | 2 | eleq1d 2826 | . 2 ⊢ (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels )) |
| 4 | 1, 3 | rabeqel 38255 | 1 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≀ ccoss 38182 Rels crels 38184 CnvRefRels ccnvrefrels 38190 FunsALTV cfunsALTV 38212 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-in 3958 df-br 5144 df-opab 5206 df-coss 38412 df-funss 38681 df-funsALTV 38682 |
| This theorem is referenced by: elfunsALTV2 38694 elfunsALTV3 38695 elfunsALTV4 38696 elfunsALTV5 38697 elfunsALTVfunALTV 38698 |
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