Theorem ereq1 8698

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 5763	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
2		dmeq 5893	. . . 4 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
3	2	eqeq1d 2765	. . 3 ⊢ (𝑅 = 𝑆 → (dom 𝑅 = 𝐴 ↔ dom 𝑆 = 𝐴))
4		cnveq 5859	. . . . . 6 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
5		coeq1 5843	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑅))
6		coeq2 5844	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑆 ∘ 𝑅) = (𝑆 ∘ 𝑆))
7	5, 6	eqtrd 2798	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆))
8	4, 7	uneq12d 4123	. . . . 5 ⊢ (𝑅 = 𝑆 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) = (◡𝑆 ∪ (𝑆 ∘ 𝑆)))
9	8	sseq1d 3968	. . . 4 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅))
10		sseq2 3963	. . . 4 ⊢ (𝑅 = 𝑆 → ((◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
11	9, 10	bitrd 282	. . 3 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
12	1, 3, 11	3anbi123d 1464	. 2 ⊢ (𝑅 = 𝑆 → ((Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)))
13		df-er 8690	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
14		df-er 8690	. 2 ⊢ (𝑆 Er 𝐴 ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
15	12, 13, 14	3bitr4g 317	1 ⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∪ cun 3903   ⊆ wss 3905  ^◡ccnv 5660  dom cdm 5661   ∘ ccom 5665  Rel wrel 5666   Er wer 8687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-er 8690

This theorem is referenced by:  riiner  8784  efglem  19781  efger  19783  efgrelexlemb  19815  efgcpbllemb  19820  frgpuplem  19837  tgjustf  28742  qtophaus  34226  pstmxmet  34287  prjspner  43351