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Theorem ereq1 8641

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 5845	. . . 4 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
3	2	eqeq1d 2741	. . 3 ⊢ (𝑅 = 𝑆 → (dom 𝑅 = 𝐴 ↔ dom 𝑆 = 𝐴))
4		cnveq 5815	. . . . . 6 ⊢ (𝑅 = 𝑆 → ^◡𝑅 = ^◡𝑆)
5		coeq1 5799	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑅))
6		coeq2 5800	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑆 ∘ 𝑅) = (𝑆 ∘ 𝑆))
7	5, 6	eqtrd 2774	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆))
8	4, 7	uneq12d 4099	. . . . 5 ⊢ (𝑅 = 𝑆 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) = (^◡𝑆 ∪ (𝑆 ∘ 𝑆)))
9	8	sseq1d 3946	. . . 4 ⊢ (𝑅 = 𝑆 → ((^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅))
10		sseq2 3941	. . . 4 ⊢ (𝑅 = 𝑆 → ((^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
11	9, 10	bitrd 280	. . 3 ⊢ (𝑅 = 𝑆 → ((^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
12	1, 3, 11	3anbi123d 1444	. 2 ⊢ (𝑅 = 𝑆 → ((Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)))
13		df-er 8633	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
14		df-er 8633	. 2 ⊢ (𝑆 Er 𝐴 ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
15	12, 13, 14	3bitr4g 315	1 ⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1092 = wceq 1547 ∪ cun 3881 ⊆ wss 3883 ^◡ccnv 5617 dom cdm 5618 ∘ ccom 5622 Rel wrel 5623 Er wer 8630

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-ext 2711

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-sb 2074 df-clab 2718 df-cleq 2731 df-clel 2814 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-ss 3900 df-nul 4262 df-if 4455 df-sn 4556 df-pr 4558 df-op 4562 df-br 5073 df-opab 5135 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-er 8633

This theorem is referenced by: riiner 8727 efglem 19682 efger 19684 efgrelexlemb 19716 efgcpbllemb 19721 frgpuplem 19738 tgjustf 28559 qtophaus 34020 pstmxmet 34081 prjspner 43069

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