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Mirrors > Home > MPE Home > Th. List > Mathboxes > pet2 | Structured version Visualization version GIF version |
Description: Partition-Equivalence Theorem, with general 𝑅. This theorem (together with pet 38215 and pets 38216) is the main result of my investigation into set theory, see the comment of pet 38215. (Contributed by Peter Mazsa, 24-May-2021.) (Revised by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
pet2 | ⊢ (( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelqseqdisj5 38197 | . 2 ⊢ (( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) → Disj (𝑅 ⋉ (◡ E ↾ 𝐴))) | |
2 | 1 | petlem 38176 | 1 ⊢ (( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 E cep 5570 ◡ccnv 5666 dom cdm 5667 ↾ cres 5669 / cqs 8699 ⋉ cxrn 37536 ≀ ccoss 37537 EqvRel weqvrel 37554 Disj wdisjALTV 37571 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-eprel 5571 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fo 6540 df-fv 6542 df-1st 7969 df-2nd 7970 df-ec 8702 df-qs 8706 df-xrn 37735 df-coss 37775 df-refrel 37876 df-cnvrefrel 37891 df-symrel 37908 df-trrel 37938 df-eqvrel 37949 df-funALTV 38046 df-disjALTV 38069 df-eldisj 38071 |
This theorem is referenced by: pet 38215 |
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