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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 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 )) |
Colors of variables: wff setvar class |
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 |
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