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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpets | Structured version Visualization version GIF version |
Description: Member Partition-Equivalence Theorem in its shortest possible form: it shows that member partitions and comember equivalence relations are literally the same. Cf. pet 38225, the Partition-Equivalence Theorem, with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.) |
Ref | Expression |
---|---|
mpets | ⊢ MembParts = CoMembErs |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpets2 38215 | . . . 4 ⊢ (𝑎 ∈ V → ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎)) | |
2 | 1 | elv 3472 | . . 3 ⊢ ((◡ E ↾ 𝑎) Parts 𝑎 ↔ ≀ (◡ E ↾ 𝑎) Ers 𝑎) |
3 | 2 | abbii 2794 | . 2 ⊢ {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎} = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎} |
4 | df-membparts 38141 | . 2 ⊢ MembParts = {𝑎 ∣ (◡ E ↾ 𝑎) Parts 𝑎} | |
5 | df-comembers 38039 | . 2 ⊢ CoMembErs = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) Ers 𝑎} | |
6 | 3, 4, 5 | 3eqtr4i 2762 | 1 ⊢ MembParts = CoMembErs |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 {cab 2701 Vcvv 3466 class class class wbr 5139 E cep 5570 ◡ccnv 5666 ↾ cres 5669 ≀ ccoss 37547 Ers cers 37572 CoMembErs ccomembers 37574 Parts cparts 37585 MembParts cmembparts 37587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-id 5565 df-eprel 5571 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ec 8702 df-qs 8706 df-coss 37785 df-coels 37786 df-rels 37859 df-ssr 37872 df-refs 37884 df-refrels 37885 df-refrel 37886 df-cnvrefs 37899 df-cnvrefrels 37900 df-cnvrefrel 37901 df-syms 37916 df-symrels 37917 df-symrel 37918 df-trs 37946 df-trrels 37947 df-trrel 37948 df-eqvrels 37958 df-eqvrel 37959 df-coeleqvrel 37961 df-dmqss 38012 df-dmqs 38013 df-ers 38037 df-erALTV 38038 df-comembers 38039 df-comember 38040 df-funALTV 38056 df-disjss 38077 df-disjs 38078 df-disjALTV 38079 df-eldisj 38081 df-parts 38139 df-part 38140 df-membparts 38141 df-membpart 38142 |
This theorem is referenced by: (None) |
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