eqvrelqseqdisj3 - Mathbox for Peter Mazsa

	Mathbox for Peter Mazsa		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelqseqdisj3		Structured version Visualization version GIF version

Theorem eqvrelqseqdisj3 38813

Description: Implication of eqvreldisj3 38808, lemma for the Member Partition Equivalence Theorem mpet3 38818. (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 38808	. . 3 ⊢ ( EqvRel 𝑅 → Disj (^◡ E ↾ (𝐵 / 𝑅)))
2	1	adantr 480	. 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (^◡ E ↾ (𝐵 / 𝑅)))
3		reseq2 5995	. . . 4 ⊢ ((𝐵 / 𝑅) = 𝐴 → (^◡ E ↾ (𝐵 / 𝑅)) = (^◡ E ↾ 𝐴))
4	3	disjeqd 38718	. . 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 1537 E cep 5588 ^◡ccnv 5688 ↾ cres 5691 / cqs 8743 EqvRel weqvrel 38179 Disj wdisjALTV 38196

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-id 5583 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 df-qs 8750 df-coss 38393 df-refrel 38494 df-cnvrefrel 38509 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 df-funALTV 38664 df-disjALTV 38687 df-eldisj 38689

This theorem is referenced by: eqvrelqseqdisj4 38814 eqvrelqseqdisj5 38815 mpet3 38818 cpet2 38819

W3C validator