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Mirrors > Home > MPE Home > Th. List > Mathboxes > elcoeleqvrels | Structured version Visualization version GIF version |
Description: Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.) |
Ref | Expression |
---|---|
elcoeleqvrels | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reseq2 5841 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴)) | |
2 | 1 | cosseqd 35707 | . . 3 ⊢ (𝑎 = 𝐴 → ≀ (◡ E ↾ 𝑎) = ≀ (◡ E ↾ 𝐴)) |
3 | 2 | eleq1d 2896 | . 2 ⊢ (𝑎 = 𝐴 → ( ≀ (◡ E ↾ 𝑎) ∈ EqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) |
4 | df-coeleqvrels 35855 | . 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | |
5 | 3, 4 | elab2g 3664 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 E cep 5457 ◡ccnv 5547 ↾ cres 5550 ≀ ccoss 35487 EqvRels ceqvrels 35503 CoElEqvRels ccoeleqvrels 35505 |
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-in 3936 df-br 5060 df-opab 5122 df-xp 5554 df-res 5560 df-coss 35693 df-coeleqvrels 35855 |
This theorem is referenced by: elcoeleqvrelsrel 35865 |
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