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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 37705 (as opposed to
petincnvepres 37714). 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 37317.
This theorem (together with pets 37717 and pet2 37715) is the main result of my investigation into set theory. It is no more general than the conventional Member Partition-Equivalence Theorem mpet 37704, mpet2 37705 and mpet3 37701 (because you cannot set 𝑅 in this theorem in such a way that you get mpet2 37705), 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 37705 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 37715 | . 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 37634 | . 2 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴)) | |
3 | dferALTV2 37533 | . 2 ⊢ ( ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) / ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴)) | |
4 | 1, 2, 3 | 3bitr4i 302 | 1 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 E cep 5579 ◡ccnv 5675 dom cdm 5676 ↾ cres 5678 / cqs 8701 ⋉ cxrn 37037 ≀ ccoss 37038 EqvRel weqvrel 37055 ErALTV werALTV 37064 Disj wdisjALTV 37072 Part wpart 37077 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7974 df-2nd 7975 df-ec 8704 df-qs 8708 df-xrn 37236 df-coss 37276 df-refrel 37377 df-cnvrefrel 37392 df-symrel 37409 df-trrel 37439 df-eqvrel 37450 df-dmqs 37504 df-erALTV 37529 df-funALTV 37547 df-disjALTV 37570 df-eldisj 37572 df-part 37631 |
This theorem is referenced by: pets 37717 |
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