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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 38312. (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 38312 | . 2 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴) | |
2 | cnvepresex 37806 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (◡ E ↾ 𝐴) ∈ V) | |
3 | brpartspart 38245 | . . . 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 37893 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ≀ (◡ E ↾ 𝐴) ∈ V) | |
6 | brerser 38149 | . . . 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 205 ∈ wcel 2099 Vcvv 3471 class class class wbr 5148 E cep 5581 ◡ccnv 5677 ↾ cres 5680 ≀ ccoss 37648 Ers cers 37673 ErALTV werALTV 37674 Parts cparts 37686 Part wpart 37687 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-id 5576 df-eprel 5582 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8727 df-qs 8731 df-coss 37883 df-coels 37884 df-rels 37957 df-ssr 37970 df-refs 37982 df-refrels 37983 df-refrel 37984 df-cnvrefs 37997 df-cnvrefrels 37998 df-cnvrefrel 37999 df-syms 38014 df-symrels 38015 df-symrel 38016 df-trs 38044 df-trrels 38045 df-trrel 38046 df-eqvrels 38056 df-eqvrel 38057 df-coeleqvrel 38059 df-dmqss 38110 df-dmqs 38111 df-ers 38135 df-erALTV 38136 df-comember 38138 df-funALTV 38154 df-disjss 38175 df-disjs 38176 df-disjALTV 38177 df-eldisj 38179 df-parts 38237 df-part 38238 df-membpart 38240 |
This theorem is referenced by: mpets 38314 |
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