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Mirrors > Home > MPE Home > Th. List > Mathboxes > elcoeleqvrelsrel | Structured version Visualization version GIF version |
Description: For sets, being an element of the class of coelement equivalence relations is equivalent to satisfying the coelement equivalence relation predicate. (Contributed by Peter Mazsa, 24-Jul-2023.) |
Ref | Expression |
---|---|
elcoeleqvrelsrel | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elcoeleqvrels 38004 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) | |
2 | 1cosscnvepresex 37830 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V) | |
3 | eleqvrelsrel 38003 | . . . 4 ⊢ ( ≀ (◡ E ↾ 𝐴) ∈ V → ( ≀ (◡ E ↾ 𝐴) ∈ EqvRels ↔ EqvRel ≀ (◡ E ↾ 𝐴))) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ( ≀ (◡ E ↾ 𝐴) ∈ EqvRels ↔ EqvRel ≀ (◡ E ↾ 𝐴))) |
5 | 1, 4 | bitrd 279 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ EqvRel ≀ (◡ E ↾ 𝐴))) |
6 | df-coeleqvrel 37996 | . 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
7 | 5, 6 | bitr4di 289 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2099 Vcvv 3469 E cep 5575 ◡ccnv 5671 ↾ cres 5674 ≀ ccoss 37583 EqvRels ceqvrels 37599 EqvRel weqvrel 37600 CoElEqvRels ccoeleqvrels 37601 CoElEqvRel wcoeleqvrel 37602 |
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-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-id 5570 df-eprel 5576 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-coss 37820 df-rels 37894 df-ssr 37907 df-refs 37919 df-refrels 37920 df-refrel 37921 df-syms 37951 df-symrels 37952 df-symrel 37953 df-trs 37981 df-trrels 37982 df-trrel 37983 df-eqvrels 37993 df-eqvrel 37994 df-coeleqvrels 37995 df-coeleqvrel 37996 |
This theorem is referenced by: (None) |
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