eqvrelqseqdisj3 - Mathbox for Peter Mazsa

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Theorem eqvrelqseqdisj3 37051

Description: Implication of eqvreldisj3 37046, lemma for the Member Partition Equivalence Theorem mpet3 37056. (Contributed by Peter Mazsa, 27-Oct-2020.) (Revised by Peter Mazsa, 24-Sep-2021.)

**Assertion**
Ref	Expression
eqvrelqseqdisj3	⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (^◡ E ↾ 𝐴))

**Proof of Theorem eqvrelqseqdisj3**
Step	Hyp	Ref	Expression
1		eqvreldisj3 37046	. . 3 ⊢ ( EqvRel 𝑅 → Disj (^◡ E ↾ (𝐵 / 𝑅)))
2	1	adantr 482	. 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (^◡ E ↾ (𝐵 / 𝑅)))
3		reseq2 5898	. . . 4 ⊢ ((𝐵 / 𝑅) = 𝐴 → (^◡ E ↾ (𝐵 / 𝑅)) = (^◡ E ↾ 𝐴))
4	3	disjeqd 36956	. . 3 ⊢ ((𝐵 / 𝑅) = 𝐴 → ( Disj (^◡ E ↾ (𝐵 / 𝑅)) ↔ Disj (^◡ E ↾ 𝐴)))
5	4	adantl 483	. 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → ( Disj (^◡ E ↾ (𝐵 / 𝑅)) ↔ Disj (^◡ E ↾ 𝐴)))
6	2, 5	mpbid 231	1 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (^◡ E ↾ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1539 E cep 5505 ^◡ccnv 5599 ↾ cres 5602 / cqs 8528 EqvRel weqvrel 36404 Disj wdisjALTV 36421

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-funALTV 36902 df-disjALTV 36925 df-eldisj 36927

This theorem is referenced by: eqvrelqseqdisj4 37052 eqvrelqseqdisj5 37053 mpet3 37056 cpet2 37057

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