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| 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 5991 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴)) | |
| 2 | 1 | cosseqd 38430 | . . 3 ⊢ (𝑎 = 𝐴 → ≀ (◡ E ↾ 𝑎) = ≀ (◡ E ↾ 𝐴)) | 
| 3 | 2 | eleq1d 2825 | . 2 ⊢ (𝑎 = 𝐴 → ( ≀ (◡ E ↾ 𝑎) ∈ EqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) | 
| 4 | df-coeleqvrels 38588 | . 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | |
| 5 | 3, 4 | elab2g 3679 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 E cep 5582 ◡ccnv 5683 ↾ cres 5686 ≀ ccoss 38183 EqvRels ceqvrels 38199 CoElEqvRels ccoeleqvrels 38201 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-in 3957 df-br 5143 df-opab 5205 df-xp 5690 df-res 5696 df-coss 38413 df-coeleqvrels 38588 | 
| This theorem is referenced by: elcoeleqvrelsrel 38598 | 
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