![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
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 38028, eqvrelcoss3 38027 and eqvrelcoss4 38029 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) |
Ref | Expression |
---|---|
dfcoeleqvrels | ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coeleqvrels 37995 | . 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | |
2 | df-coels 37821 | . . . 4 ⊢ ∼ 𝑎 = ≀ (◡ E ↾ 𝑎) | |
3 | 2 | eleq1i 2819 | . . 3 ⊢ ( ∼ 𝑎 ∈ EqvRels ↔ ≀ (◡ E ↾ 𝑎) ∈ EqvRels ) |
4 | 3 | abbii 2797 | . 2 ⊢ {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } |
5 | 1, 4 | eqtr4i 2758 | 1 ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 {cab 2704 E cep 5575 ◡ccnv 5671 ↾ cres 5674 ≀ ccoss 37583 ∼ ccoels 37584 EqvRels ceqvrels 37599 CoElEqvRels ccoeleqvrels 37601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-coels 37821 df-coeleqvrels 37995 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |