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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpet3 | Structured version Visualization version GIF version |
Description: Member Partition-Equivalence Theorem. Together with mpet 38311 mpet2 38312, mostly in its conventional cpet 38310 and cpet2 38309 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38322 with general 𝑅). (Contributed by Peter Mazsa, 4-May-2018.) (Revised by Peter Mazsa, 26-Sep-2021.) |
Ref | Expression |
---|---|
mpet3 | ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldisjn0elb 38217 | . 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴)) | |
2 | eqvrelqseqdisj3 38303 | . . 3 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) → Disj (◡ E ↾ 𝐴)) | |
3 | 2 | petlem 38284 | . 2 ⊢ (( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴)) |
4 | eqvreldmqs 38147 | . 2 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
5 | 1, 3, 4 | 3bitri 297 | 1 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∅c0 4323 ∪ cuni 4908 E cep 5581 ◡ccnv 5677 dom cdm 5678 ↾ cres 5680 / cqs 8723 ≀ ccoss 37648 ∼ ccoels 37649 EqvRel weqvrel 37665 CoElEqvRel wcoeleqvrel 37667 Disj wdisjALTV 37682 ElDisj weldisj 37684 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-eprel 5582 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8726 df-qs 8730 df-coss 37883 df-coels 37884 df-refrel 37984 df-cnvrefrel 37999 df-symrel 38016 df-trrel 38046 df-eqvrel 38057 df-coeleqvrel 38059 df-funALTV 38154 df-disjALTV 38177 df-eldisj 38179 |
This theorem is referenced by: mpet 38311 |
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