ereq1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ereq1		Structured version Visualization version GIF version

Theorem ereq1 8632

Description: Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)

**Assertion**
Ref	Expression
ereq1	⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴))

**Proof of Theorem ereq1**
Step	Hyp	Ref	Expression
1		releq 5720	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
2		dmeq 5846	. . . 4 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
3	2	eqeq1d 2731	. . 3 ⊢ (𝑅 = 𝑆 → (dom 𝑅 = 𝐴 ↔ dom 𝑆 = 𝐴))
4		cnveq 5816	. . . . . 6 ⊢ (𝑅 = 𝑆 → ^◡𝑅 = ^◡𝑆)
5		coeq1 5800	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑅))
6		coeq2 5801	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑆 ∘ 𝑅) = (𝑆 ∘ 𝑆))
7	5, 6	eqtrd 2764	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆))
8	4, 7	uneq12d 4120	. . . . 5 ⊢ (𝑅 = 𝑆 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) = (^◡𝑆 ∪ (𝑆 ∘ 𝑆)))
9	8	sseq1d 3967	. . . 4 ⊢ (𝑅 = 𝑆 → ((^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅))
10		sseq2 3962	. . . 4 ⊢ (𝑅 = 𝑆 → ((^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
11	9, 10	bitrd 279	. . 3 ⊢ (𝑅 = 𝑆 → ((^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
12	1, 3, 11	3anbi123d 1438	. 2 ⊢ (𝑅 = 𝑆 → ((Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)))
13		df-er 8625	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
14		df-er 8625	. 2 ⊢ (𝑆 Er 𝐴 ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
15	12, 13, 14	3bitr4g 314	1 ⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1540 ∪ cun 3901 ⊆ wss 3903 ^◡ccnv 5618 dom cdm 5619 ∘ ccom 5623 Rel wrel 5624 Er wer 8622

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 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-ext 2701

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2721 df-clel 2803 df-rab 3395 df-v 3438 df-dif 3906 df-un 3908 df-ss 3920 df-nul 4285 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-br 5093 df-opab 5155 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-er 8625

This theorem is referenced by: riiner 8717 efglem 19595 efger 19597 efgrelexlemb 19629 efgcpbllemb 19634 frgpuplem 19651 tgjustf 28418 qtophaus 33809 pstmxmet 33870 prjspner 42602

W3C validator