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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 5930 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴)) | |
2 | 1 | cosseqd 36828 | . . 3 ⊢ (𝑎 = 𝐴 → ≀ (◡ E ↾ 𝑎) = ≀ (◡ E ↾ 𝐴)) |
3 | 2 | eleq1d 2822 | . 2 ⊢ (𝑎 = 𝐴 → ( ≀ (◡ E ↾ 𝑎) ∈ EqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) |
4 | df-coeleqvrels 36986 | . 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | |
5 | 3, 4 | elab2g 3630 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 E cep 5534 ◡ccnv 5630 ↾ cres 5633 ≀ ccoss 36572 EqvRels ceqvrels 36588 CoElEqvRels ccoeleqvrels 36590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3406 df-in 3915 df-br 5104 df-opab 5166 df-xp 5637 df-res 5643 df-coss 36811 df-coeleqvrels 36986 |
This theorem is referenced by: elcoeleqvrelsrel 36996 |
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