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Mirrors > Home > MPE Home > Th. List > Mathboxes > petincnvepres | Structured version Visualization version GIF version |
Description: The shortest form of a partition-equivalence theorem with intersection and general 𝑅. Cf. br1cossincnvepres 36670. Cf. pet 37071. (Contributed by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
petincnvepres | ⊢ ((𝑅 ∩ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) ErALTV 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | petincnvepres2 37068 | . 2 ⊢ (( Disj (𝑅 ∩ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ∩ (◡ E ↾ 𝐴)) / (𝑅 ∩ (◡ E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) / ≀ (𝑅 ∩ (◡ E ↾ 𝐴))) = 𝐴)) | |
2 | dfpart2 36989 | . 2 ⊢ ((𝑅 ∩ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ∩ (◡ E ↾ 𝐴)) ∧ (dom (𝑅 ∩ (◡ E ↾ 𝐴)) / (𝑅 ∩ (◡ E ↾ 𝐴))) = 𝐴)) | |
3 | dferALTV2 36888 | . 2 ⊢ ( ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) / ≀ (𝑅 ∩ (◡ E ↾ 𝐴))) = 𝐴)) | |
4 | 1, 2, 3 | 3bitr4i 303 | 1 ⊢ ((𝑅 ∩ (◡ E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ (◡ E ↾ 𝐴)) ErALTV 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1539 ∩ cin 3891 E cep 5505 ◡ccnv 5599 dom cdm 5600 ↾ cres 5602 / cqs 8528 ≀ ccoss 36387 EqvRel weqvrel 36404 ErALTV werALTV 36413 Disj wdisjALTV 36421 Part wpart 36426 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3339 df-rab 3341 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-br 5082 df-opab 5144 df-id 5500 df-eprel 5506 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-ec 8531 df-qs 8535 df-coss 36631 df-refrel 36732 df-cnvrefrel 36747 df-symrel 36764 df-trrel 36794 df-eqvrel 36805 df-dmqs 36859 df-erALTV 36884 df-funALTV 36902 df-disjALTV 36925 df-eldisj 36927 df-part 36986 |
This theorem is referenced by: (None) |
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