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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 6004 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴)) | |
2 | 1 | cosseqd 38384 | . . 3 ⊢ (𝑎 = 𝐴 → ≀ (◡ E ↾ 𝑎) = ≀ (◡ E ↾ 𝐴)) |
3 | 2 | eleq1d 2829 | . 2 ⊢ (𝑎 = 𝐴 → ( ≀ (◡ E ↾ 𝑎) ∈ EqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) |
4 | df-coeleqvrels 38542 | . 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | |
5 | 3, 4 | elab2g 3696 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2108 E cep 5598 ◡ccnv 5699 ↾ cres 5702 ≀ ccoss 38135 EqvRels ceqvrels 38151 CoElEqvRels ccoeleqvrels 38153 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-rab 3444 df-in 3983 df-br 5167 df-opab 5229 df-xp 5706 df-res 5712 df-coss 38367 df-coeleqvrels 38542 |
This theorem is referenced by: elcoeleqvrelsrel 38552 |
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