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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 35888, eqvrelcoss3 35887 and eqvrelcoss4 35889 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.) |
Ref | Expression |
---|---|
dfcoeleqvrels | ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coeleqvrels 35855 | . 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } | |
2 | df-coels 35694 | . . . 4 ⊢ ∼ 𝑎 = ≀ (◡ E ↾ 𝑎) | |
3 | 2 | eleq1i 2902 | . . 3 ⊢ ( ∼ 𝑎 ∈ EqvRels ↔ ≀ (◡ E ↾ 𝑎) ∈ EqvRels ) |
4 | 3 | abbii 2885 | . 2 ⊢ {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels } |
5 | 1, 4 | eqtr4i 2846 | 1 ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 {cab 2798 E cep 5457 ◡ccnv 5547 ↾ cres 5550 ≀ ccoss 35487 ∼ ccoels 35488 EqvRels ceqvrels 35503 CoElEqvRels ccoeleqvrels 35505 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-coels 35694 df-coeleqvrels 35855 |
This theorem is referenced by: (None) |
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