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| Description: Partition-Equivalence
Theorem with general 𝑅 while preserving the
     restricted converse epsilon relation of mpet2 38841 (as opposed to
     petincnvepres 38850).  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 38453. This theorem (together with pets 38853 and pet2 38851) is the main result of my investigation into set theory. It is no more general than the conventional Member Partition-Equivalence Theorem mpet 38840, mpet2 38841 and mpet3 38837 (because you cannot set 𝑅 in this theorem in such a way that you get mpet2 38841), 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 38841 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 38851 | . 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 38770 | . 2 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ⋉ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ (◡ E ↾ 𝐴)) / (𝑅 ⋉ (◡ E ↾ 𝐴))) = 𝐴)) | |
| 3 | dferALTV2 38669 | . 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 206 ∧ wa 395 = wceq 1540 E cep 5583 ◡ccnv 5684 dom cdm 5685 ↾ cres 5687 / cqs 8744 ⋉ cxrn 38181 ≀ ccoss 38182 EqvRel weqvrel 38199 ErALTV werALTV 38208 Disj wdisjALTV 38216 Part wpart 38221 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-ec 8747 df-qs 8751 df-xrn 38372 df-coss 38412 df-refrel 38513 df-cnvrefrel 38528 df-symrel 38545 df-trrel 38575 df-eqvrel 38586 df-dmqs 38640 df-erALTV 38665 df-funALTV 38683 df-disjALTV 38706 df-eldisj 38708 df-part 38767 | 
| This theorem is referenced by: pets 38853 | 
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