Theorem ereq1 6520

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 4693	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
2		dmeq 4811	. . . 4 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
3	2	eqeq1d 2179	. . 3 ⊢ (𝑅 = 𝑆 → (dom 𝑅 = 𝐴 ↔ dom 𝑆 = 𝐴))
4		cnveq 4785	. . . . . 6 ⊢ (𝑅 = 𝑆 → ◡𝑅 = ◡𝑆)
5		coeq1 4768	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑅))
6		coeq2 4769	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑆 ∘ 𝑅) = (𝑆 ∘ 𝑆))
7	5, 6	eqtrd 2203	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆))
8	4, 7	uneq12d 3282	. . . . 5 ⊢ (𝑅 = 𝑆 → (◡𝑅 ∪ (𝑅 ∘ 𝑅)) = (◡𝑆 ∪ (𝑆 ∘ 𝑆)))
9	8	sseq1d 3176	. . . 4 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅))
10		sseq2 3171	. . . 4 ⊢ (𝑅 = 𝑆 → ((◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
11	9, 10	bitrd 187	. . 3 ⊢ (𝑅 = 𝑆 → ((◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
12	1, 3, 11	3anbi123d 1307	. 2 ⊢ (𝑅 = 𝑆 → ((Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)))
13		df-er 6513	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
14		df-er 6513	. 2 ⊢ (𝑆 Er 𝐴 ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
15	12, 13, 14	3bitr4g 222	1 ⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴))

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 973   = wceq 1348   ∪ cun 3119   ⊆ wss 3121  ^◡ccnv 4610  dom cdm 4611   ∘ ccom 4615  Rel wrel 4616   Er wer 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-er 6513

This theorem is referenced by:  riinerm  6586