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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpets2 | Structured version Visualization version GIF version |
Description: Member Partition-Equivalence Theorem with binary relations, cf. mpet2 38783. (Contributed by Peter Mazsa, 24-Sep-2021.) |
Ref | Expression |
---|---|
mpets2 | ⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) Parts 𝐴 ↔ ≀ (◡ E ↾ 𝐴) Ers 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpet2 38783 | . 2 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴) | |
2 | cnvepresex 38277 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | |
3 | brpartspart 38716 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (◡ E ↾ 𝐴) ∈ V) → ((◡ E ↾ 𝐴) Parts 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)) | |
4 | 2, 3 | mpdan 686 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) Parts 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)) |
5 | 1cosscnvepresex 38364 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V) | |
6 | brerser 38620 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ≀ (◡ E ↾ 𝐴) ∈ V) → ( ≀ (◡ E ↾ 𝐴) Ers 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)) | |
7 | 5, 6 | mpdan 686 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ( ≀ (◡ E ↾ 𝐴) Ers 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)) |
8 | 4, 7 | bibi12d 345 | . 2 ⊢ (𝐴 ∈ 𝑉 → (((◡ E ↾ 𝐴) Parts 𝐴 ↔ ≀ (◡ E ↾ 𝐴) Ers 𝐴) ↔ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴))) |
9 | 1, 8 | mpbiri 258 | 1 ⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) Parts 𝐴 ↔ ≀ (◡ E ↾ 𝐴) Ers 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2104 Vcvv 3477 class class class wbr 5149 E cep 5581 ◡ccnv 5682 ↾ cres 5685 ≀ ccoss 38122 Ers cers 38147 ErALTV werALTV 38148 Parts cparts 38160 Part wpart 38161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-id 5576 df-eprel 5582 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-ec 8740 df-qs 8744 df-coss 38354 df-coels 38355 df-rels 38428 df-ssr 38441 df-refs 38453 df-refrels 38454 df-refrel 38455 df-cnvrefs 38468 df-cnvrefrels 38469 df-cnvrefrel 38470 df-syms 38485 df-symrels 38486 df-symrel 38487 df-trs 38515 df-trrels 38516 df-trrel 38517 df-eqvrels 38527 df-eqvrel 38528 df-coeleqvrel 38530 df-dmqss 38581 df-dmqs 38582 df-ers 38606 df-erALTV 38607 df-comember 38609 df-funALTV 38625 df-disjss 38646 df-disjs 38647 df-disjALTV 38648 df-eldisj 38650 df-parts 38708 df-part 38709 df-membpart 38711 |
This theorem is referenced by: mpets 38785 |
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