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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pet | Structured version Visualization version GIF version |
Description: Partition-Equivalence
Theorem with general 𝑅 while preserving the
restricted converse epsilon relation of mpet2 37710 (as opposed to
petincnvepres 37719). A class is a partition by a range
Cartesian product
with general 𝑅 and the restricted converse element
class if and only
if the cosets by the range Cartesian product are in an equivalence
relation on it. Cf. br1cossxrncnvepres 37322.
This theorem (together with pets 37722 and pet2 37720) is the main result of my investigation into set theory. It is no more general than the conventional Member Partition-Equivalence Theorem mpet 37709, mpet2 37710 and mpet3 37706 (because you cannot set 𝑅 in this theorem in such a way that you get mpet2 37710), i.e., it is not the hypothetical General Partition-Equivalence Theorem gpet ⊢ (𝑅 Part 𝐴 ↔ ≀ 𝑅 ErALTV 𝐴), but this one has a general part that mpet2 37710 lacks: 𝑅, which is sufficient for my future application of set theory, for my purpose outside of set theory. (Contributed by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
pet | ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pet2 37720 | . 2 ⊢ (( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴)) | |
2 | dfpart2 37639 | . 2 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴)) | |
3 | dferALTV2 37538 | . 2 ⊢ ( ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴)) | |
4 | 1, 2, 3 | 3bitr4i 303 | 1 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 E cep 5580 ◡ccnv 5676 dom cdm 5677 ↾ cres 5679 / cqs 8702 ⋉ cxrn 37042 ≀ ccoss 37043 EqvRel weqvrel 37060 ErALTV werALTV 37069 Disj wdisjALTV 37077 Part wpart 37082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-eprel 5581 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fo 6550 df-fv 6552 df-1st 7975 df-2nd 7976 df-ec 8705 df-qs 8709 df-xrn 37241 df-coss 37281 df-refrel 37382 df-cnvrefrel 37397 df-symrel 37414 df-trrel 37444 df-eqvrel 37455 df-dmqs 37509 df-erALTV 37534 df-funALTV 37552 df-disjALTV 37575 df-eldisj 37577 df-part 37636 |
This theorem is referenced by: pets 37722 |
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