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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 38620, eqvrelcoss3 38619 and eqvrelcoss4 38621 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) | 
| Ref | Expression | 
|---|---|
| dfcoeleqvrels | ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-coeleqvrels 38587 | . 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | |
| 2 | df-coels 38413 | . . . 4 ⊢ ∼ 𝑎 = ≀ (◡ E ↾ 𝑎) | |
| 3 | 2 | eleq1i 2832 | . . 3 ⊢ ( ∼ 𝑎 ∈ EqvRels ↔ ≀ (◡ E ↾ 𝑎) ∈ EqvRels ) | 
| 4 | 3 | abbii 2809 | . 2 ⊢ {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | 
| 5 | 1, 4 | eqtr4i 2768 | 1 ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 {cab 2714 E cep 5583 ◡ccnv 5684 ↾ cres 5687 ≀ ccoss 38182 ∼ ccoels 38183 EqvRels ceqvrels 38198 CoElEqvRels ccoeleqvrels 38200 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-coels 38413 df-coeleqvrels 38587 | 
| This theorem is referenced by: (None) | 
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