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 38240

Description: Implication of eqvreldisj3 38235, lemma for the Member Partition Equivalence Theorem mpet3 38245. (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 38235	. . 3 ⊢ ( EqvRel 𝑅 → Disj (^◡ E ↾ (𝐵 / 𝑅)))
2	1	adantr 480	. 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (^◡ E ↾ (𝐵 / 𝑅)))
3		reseq2 5974	. . . 4 ⊢ ((𝐵 / 𝑅) = 𝐴 → (^◡ E ↾ (𝐵 / 𝑅)) = (^◡ E ↾ 𝐴))
4	3	disjeqd 38145	. . 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 231	1 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (^◡ E ↾ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 E cep 5575 ^◡ccnv 5671 ↾ cres 5674 / cqs 8717 EqvRel weqvrel 37600 Disj wdisjALTV 37617

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-eprel 5576 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ec 8720 df-qs 8724 df-coss 37820 df-refrel 37921 df-cnvrefrel 37936 df-symrel 37953 df-trrel 37983 df-eqvrel 37994 df-funALTV 38091 df-disjALTV 38114 df-eldisj 38116

This theorem is referenced by: eqvrelqseqdisj4 38241 eqvrelqseqdisj5 38242 mpet3 38245 cpet2 38246

W3C validator