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Theorem ereq1 6685

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 4800	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
2		dmeq 4922	. . . 4 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
3	2	eqeq1d 2238	. . 3 ⊢ (𝑅 = 𝑆 → (dom 𝑅 = 𝐴 ↔ dom 𝑆 = 𝐴))
4		cnveq 4895	. . . . . 6 ⊢ (𝑅 = 𝑆 → ^◡𝑅 = ^◡𝑆)
5		coeq1 4878	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑅))
6		coeq2 4879	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑆 ∘ 𝑅) = (𝑆 ∘ 𝑆))
7	5, 6	eqtrd 2262	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆))
8	4, 7	uneq12d 3359	. . . . 5 ⊢ (𝑅 = 𝑆 → (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) = (^◡𝑆 ∪ (𝑆 ∘ 𝑆)))
9	8	sseq1d 3253	. . . 4 ⊢ (𝑅 = 𝑆 → ((^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅))
10		sseq2 3248	. . . 4 ⊢ (𝑅 = 𝑆 → ((^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
11	9, 10	bitrd 188	. . 3 ⊢ (𝑅 = 𝑆 → ((^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
12	1, 3, 11	3anbi123d 1346	. 2 ⊢ (𝑅 = 𝑆 → ((Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)))
13		df-er 6678	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
14		df-er 6678	. 2 ⊢ (𝑆 Er 𝐴 ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
15	12, 13, 14	3bitr4g 223	1 ⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 1002 = wceq 1395 ∪ cun 3195 ⊆ wss 3197 ^◡ccnv 4717 dom cdm 4718 ∘ ccom 4722 Rel wrel 4723 Er wer 6675

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-ext 2211

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-v 2801 df-un 3201 df-in 3203 df-ss 3210 df-sn 3672 df-pr 3673 df-op 3675 df-br 4083 df-opab 4145 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-er 6678

This theorem is referenced by: riinerm 6753

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