Theorem ereq1 8704

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 5765	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
2		dmeq 5895	. . . 4 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
3	2	eqeq1d 2767	. . 3 ⊢ (𝑅 = 𝑆 → (dom 𝑅 = 𝐴 ↔ dom 𝑆 = 𝐴))
4		cnveq 5861	. . . . . 6 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
5		coeq1 5845	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑅))
6		coeq2 5846	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑆 ∘ 𝑅) = (𝑆 ∘ 𝑆))
7	5, 6	eqtrd 2800	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆))
8	4, 7	uneq12d 4123	. . . . 5 ⊢ (𝑅 = 𝑆 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) = (◡𝑆 ∪ (𝑆 ∘ 𝑆)))
9	8	sseq1d 3969	. . . 4 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅))
10		sseq2 3964	. . . 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 8696	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
14		df-er 8696	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1570   ∪ cun 3904   ⊆ wss 3906  ^◡ccnv 5662  dom cdm 5663   ∘ ccom 5667  Rel wrel 5668   Er wer 8693

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-er 8696

This theorem is used by:  riiner  8790  efglem  19810  efger  19812  efgrelexlemb  19844  efgcpbllemb  19849  frgpuplem  19866  tgjustf  28773  qtophaus  34266  pstmxmet  34327  prjspner  43384