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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 35931 | . 2 ⊢ FunsALTV = {𝑥 ∈ Rels ∣ ≀ 𝑥 ∈ CnvRefRels } | |
2 | cosseq 35686 | . . 3 ⊢ (𝑥 = 𝐹 → ≀ 𝑥 = ≀ 𝐹) | |
3 | 2 | eleq1d 2897 | . 2 ⊢ (𝑥 = 𝐹 → ( ≀ 𝑥 ∈ CnvRefRels ↔ ≀ 𝐹 ∈ CnvRefRels )) |
4 | 1, 3 | rabeqel 35531 | 1 ⊢ (𝐹 ∈ FunsALTV ↔ ( ≀ 𝐹 ∈ CnvRefRels ∧ 𝐹 ∈ Rels )) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≀ ccoss 35468 Rels crels 35470 CnvRefRels ccnvrefrels 35476 FunsALTV cfunsALTV 35498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3943 df-br 5067 df-opab 5129 df-coss 35674 df-funss 35928 df-funsALTV 35929 |
This theorem is referenced by: elfunsALTV2 35941 elfunsALTV3 35942 elfunsALTV4 35943 elfunsALTV5 35944 elfunsALTVfunALTV 35945 |
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