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Mirrors > Home > MPE Home > Th. List > Mathboxes > pets | Structured version Visualization version GIF version |
Description: Partition-Equivalence Theorem with general 𝑅, with binary relations. This theorem (together with pet 37526 and pet2 37525) is the main result of my investigation into set theory, cf. the comment of pet 37526. (Contributed by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
pets | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pet 37526 | . 2 ⊢ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴) | |
2 | xrncnvepresex 37083 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) | |
3 | brpartspart 37448 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ (𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴)) | |
4 | 2, 3 | syldan 591 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ (𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴)) |
5 | 1cossxrncnvepresex 37097 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) | |
6 | brerser 37352 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ∈ V) → ( ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴)) | |
7 | 5, 6 | syldan 591 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ( ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴)) |
8 | 4, 7 | bibi12d 345 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → (((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴) ↔ ((𝑅 ⋉ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) ErALTV 𝐴))) |
9 | 1, 8 | mpbiri 257 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊) → ((𝑅 ⋉ (◡ E ↾ 𝐴)) Parts 𝐴 ↔ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴)) Ers 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3473 class class class wbr 5141 E cep 5572 ◡ccnv 5668 ↾ cres 5671 ⋉ cxrn 36847 ≀ ccoss 36848 Ers cers 36873 ErALTV werALTV 36874 Parts cparts 36886 Part wpart 36887 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fo 6538 df-fv 6540 df-1st 7957 df-2nd 7958 df-ec 8688 df-qs 8692 df-xrn 37046 df-coss 37086 df-rels 37160 df-ssr 37173 df-refs 37185 df-refrels 37186 df-refrel 37187 df-cnvrefs 37200 df-cnvrefrels 37201 df-cnvrefrel 37202 df-syms 37217 df-symrels 37218 df-symrel 37219 df-trs 37247 df-trrels 37248 df-trrel 37249 df-eqvrels 37259 df-eqvrel 37260 df-dmqss 37313 df-dmqs 37314 df-ers 37338 df-erALTV 37339 df-funALTV 37357 df-disjss 37378 df-disjs 37379 df-disjALTV 37380 df-eldisj 37382 df-parts 37440 df-part 37441 |
This theorem is referenced by: (None) |
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