ereq1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ereq1		Structured version Visualization version GIF version

Theorem ereq1 8279

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 5615	. . 3 ⊢ (𝑅 = 𝑆 → (Rel 𝑅 ↔ Rel 𝑆))
2		dmeq 5736	. . . 4 ⊢ (𝑅 = 𝑆 → dom 𝑅 = dom 𝑆)
3	2	eqeq1d 2800	. . 3 ⊢ (𝑅 = 𝑆 → (dom 𝑅 = 𝐴 ↔ dom 𝑆 = 𝐴))
4		cnveq 5708	. . . . . 6 ⊢ (𝑅 = 𝑆 → ^◡𝑅 = ^◡𝑆)
5		coeq1 5692	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑅))
6		coeq2 5693	. . . . . . 7 ⊢ (𝑅 = 𝑆 → (𝑆 ∘ 𝑅) = (𝑆 ∘ 𝑆))
7	5, 6	eqtrd 2833	. . . . . 6 ⊢ (𝑅 = 𝑆 → (𝑅 ∘ 𝑅) = (𝑆 ∘ 𝑆))
8	4, 7	uneq12d 4091	. . . . 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 282	. . 3 ⊢ (𝑅 = 𝑆 → ((^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅 ↔ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆))
12	1, 3, 11	3anbi123d 1433	. 2 ⊢ (𝑅 = 𝑆 → ((Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅) ↔ (Rel 𝑆 ∧ dom 𝑆 = 𝐴 ∧ (^◡𝑆 ∪ (𝑆 ∘ 𝑆)) ⊆ 𝑆)))
13		df-er 8272	. 2 ⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (^◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅))
14		df-er 8272	. 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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∪ cun 3879 ⊆ wss 3881 ^◡ccnv 5518 dom cdm 5519 ∘ ccom 5523 Rel wrel 5524 Er wer 8269

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-er 8272

This theorem is referenced by: riiner 8353 efglem 18834 efger 18836 efgrelexlemb 18868 efgcpbllemb 18873 frgpuplem 18890 tgjustf 26267 qtophaus 31189 pstmxmet 31250

W3C validator