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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 38819. (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 38819 | . 2 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴) | |
| 2 | cnvepresex 38313 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | |
| 3 | brpartspart 38752 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (◡ E ↾ 𝐴) ∈ V) → ((◡ E ↾ 𝐴) Parts 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)) | |
| 4 | 2, 3 | mpdan 687 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((◡ E ↾ 𝐴) Parts 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)) |
| 5 | 1cosscnvepresex 38400 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V) | |
| 6 | brerser 38656 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ≀ (◡ E ↾ 𝐴) ∈ V) → ( ≀ (◡ E ↾ 𝐴) Ers 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)) | |
| 7 | 5, 6 | mpdan 687 | . . 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 2108 Vcvv 3479 class class class wbr 5141 E cep 5581 ◡ccnv 5682 ↾ cres 5685 ≀ ccoss 38160 Ers cers 38185 ErALTV werALTV 38186 Parts cparts 38198 Part wpart 38199 |
| 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 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
| 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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 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 8743 df-qs 8747 df-coss 38390 df-coels 38391 df-rels 38464 df-ssr 38477 df-refs 38489 df-refrels 38490 df-refrel 38491 df-cnvrefs 38504 df-cnvrefrels 38505 df-cnvrefrel 38506 df-syms 38521 df-symrels 38522 df-symrel 38523 df-trs 38551 df-trrels 38552 df-trrel 38553 df-eqvrels 38563 df-eqvrel 38564 df-coeleqvrel 38566 df-dmqss 38617 df-dmqs 38618 df-ers 38642 df-erALTV 38643 df-comember 38645 df-funALTV 38661 df-disjss 38682 df-disjs 38683 df-disjALTV 38684 df-eldisj 38686 df-parts 38744 df-part 38745 df-membpart 38747 |
| This theorem is referenced by: mpets 38821 |
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