eqvrelqseqdisj3 - Mathbox for Peter Mazsa

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Theorem eqvrelqseqdisj3 38787

Description: Implication of eqvreldisj3 38782, lemma for the Member Partition Equivalence Theorem mpet3 38792. (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 38782	. . 3 ⊢ ( EqvRel 𝑅 → Disj (^◡ E ↾ (𝐵 / 𝑅)))
2	1	adantr 480	. 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (^◡ E ↾ (𝐵 / 𝑅)))
3		reseq2 6004	. . . 4 ⊢ ((𝐵 / 𝑅) = 𝐴 → (^◡ E ↾ (𝐵 / 𝑅)) = (^◡ E ↾ 𝐴))
4	3	disjeqd 38692	. . 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 5598 ^◡ccnv 5699 ↾ cres 5702 / cqs 8762 EqvRel weqvrel 38152 Disj wdisjALTV 38169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-id 5593 df-eprel 5599 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-ec 8765 df-qs 8769 df-coss 38367 df-refrel 38468 df-cnvrefrel 38483 df-symrel 38500 df-trrel 38530 df-eqvrel 38541 df-funALTV 38638 df-disjALTV 38661 df-eldisj 38663

This theorem is referenced by: eqvrelqseqdisj4 38788 eqvrelqseqdisj5 38789 mpet3 38792 cpet2 38793

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