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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 36402 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels )) | |
2 | 1cosscnvepresex 36238 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V) | |
3 | eleqvrelsrel 36401 | . . . 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 282 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ EqvRel ≀ (◡ E ↾ 𝐴))) |
6 | df-coeleqvrel 36394 | . 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
7 | 5, 6 | bitr4di 292 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ CoElEqvRel 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2110 Vcvv 3401 E cep 5448 ◡ccnv 5539 ↾ cres 5542 ≀ ccoss 36027 EqvRels ceqvrels 36043 EqvRel weqvrel 36044 CoElEqvRels ccoeleqvrels 36045 CoElEqvRel wcoeleqvrel 36046 |
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-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-eprel 5449 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-coss 36231 df-rels 36297 df-ssr 36310 df-refs 36322 df-refrels 36323 df-refrel 36324 df-syms 36350 df-symrels 36351 df-symrel 36352 df-trs 36380 df-trrels 36381 df-trrel 36382 df-eqvrels 36391 df-eqvrel 36392 df-coeleqvrels 36393 df-coeleqvrel 36394 |
This theorem is referenced by: (None) |
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