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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcoeleqvrels | Structured version Visualization version GIF version |
Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 36426, eqvrelcoss3 36425 and eqvrelcoss4 36427 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) |
Ref | Expression |
---|---|
dfcoeleqvrels | ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coeleqvrels 36393 | . 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | |
2 | df-coels 36232 | . . . 4 ⊢ ∼ 𝑎 = ≀ (◡ E ↾ 𝑎) | |
3 | 2 | eleq1i 2824 | . . 3 ⊢ ( ∼ 𝑎 ∈ EqvRels ↔ ≀ (◡ E ↾ 𝑎) ∈ EqvRels ) |
4 | 3 | abbii 2804 | . 2 ⊢ {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } |
5 | 1, 4 | eqtr4i 2765 | 1 ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 {cab 2712 E cep 5448 ◡ccnv 5539 ↾ cres 5542 ≀ ccoss 36027 ∼ ccoels 36028 EqvRels ceqvrels 36043 CoElEqvRels ccoeleqvrels 36045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-coels 36232 df-coeleqvrels 36393 |
This theorem is referenced by: (None) |
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