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Mirrors > Home > MPE Home > Th. List > Mathboxes > mainer | Structured version Visualization version GIF version |
Description: The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 26-Sep-2021.) |
Ref | Expression |
---|---|
mainer | ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelqseqdisj2 38210 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴) | |
2 | eldisjim 38165 | . . . 4 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴) |
4 | n0eldmqseq 38030 | . . . . 5 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴) | |
5 | 4 | adantl 481 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴) |
6 | eldisjn0el 38187 | . . . . 5 ⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
8 | 5, 7 | mpbid 231 | . . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (∪ 𝐴 / ∼ 𝐴) = 𝐴) |
9 | 3, 8 | jca 511 | . 2 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
10 | dferALTV2 38049 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
11 | dfcomember3 38055 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
12 | 9, 10, 11 | 3imtr4i 292 | 1 ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∅c0 4317 ∪ cuni 4902 dom cdm 5669 / cqs 8701 ∼ ccoels 37555 EqvRel weqvrel 37571 CoElEqvRel wcoeleqvrel 37573 ErALTV werALTV 37580 CoMembEr wcomember 37582 ElDisj weldisj 37590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8704 df-qs 8708 df-coss 37792 df-coels 37793 df-refrel 37893 df-cnvrefrel 37908 df-symrel 37925 df-trrel 37955 df-eqvrel 37966 df-coeleqvrel 37968 df-dmqs 38020 df-erALTV 38045 df-comember 38047 df-funALTV 38063 df-disjALTV 38086 df-eldisj 38088 |
This theorem is referenced by: partimcomember 38216 fences 38225 |
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