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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 35950 | . 2 ⊢ FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels } | |
2 | cosseq 35705 | . . 3 ⊢ (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹) | |
3 | 2 | eleq1d 2896 | . 2 ⊢ (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels )) |
4 | 1, 3 | rabeqel 35550 | 1 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≀ ccoss 35487 Rels crels 35489 CnvRefRels ccnvrefrels 35495 FunsALTV cfunsALTV 35517 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-in 3936 df-br 5060 df-opab 5122 df-coss 35693 df-funss 35947 df-funsALTV 35948 |
This theorem is referenced by: elfunsALTV2 35960 elfunsALTV3 35961 elfunsALTV4 35962 elfunsALTV5 35963 elfunsALTVfunALTV 35964 |
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