eqvrelqseqdisj3 - Mathbox for Peter Mazsa

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Theorem eqvrelqseqdisj3 38833

Description: Implication of eqvreldisj3 38828, lemma for the Member Partition Equivalence Theorem mpet3 38838. (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 38828	. . 3 ⊢ ( EqvRel 𝑅 → Disj (^◡ E ↾ (𝐵 / 𝑅)))
2	1	adantr 480	. 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (^◡ E ↾ (𝐵 / 𝑅)))
3		reseq2 5991	. . . 4 ⊢ ((𝐵 / 𝑅) = 𝐴 → (^◡ E ↾ (𝐵 / 𝑅)) = (^◡ E ↾ 𝐴))
4	3	disjeqd 38738	. . 3 ⊢ ((𝐵 / 𝑅) = 𝐴 → ( Disj (^◡ E ↾ (𝐵 / 𝑅)) ↔ Disj (^◡ E ↾ 𝐴)))
5	4	adantl 481	. 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → ( Disj (^◡ E ↾ (𝐵 / 𝑅)) ↔ Disj (^◡ E ↾ 𝐴)))
6	2, 5	mpbid 232	1 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (^◡ E ↾ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 E cep 5582 ^◡ccnv 5683 ↾ cres 5686 / cqs 8745 EqvRel weqvrel 38200 Disj wdisjALTV 38217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 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-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-eprel 5583 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ec 8748 df-qs 8752 df-coss 38413 df-refrel 38514 df-cnvrefrel 38529 df-symrel 38546 df-trrel 38576 df-eqvrel 38587 df-funALTV 38684 df-disjALTV 38707 df-eldisj 38709

This theorem is referenced by: eqvrelqseqdisj4 38834 eqvrelqseqdisj5 38835 mpet3 38838 cpet2 38839

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