Theorem ereq 3206

Description: Equality theorem for equivalence predicate.

Assertion

Ref	Expression
ereq	⊢ (R = S → (Er R ↔ Er S))

Proof of Theorem ereq

Step	Hyp	Ref	Expression
1		cnveq 2513	. . . . 5 ⊢ (R = S → ◡R = ◡S)
2		coeq1 2502	. . . . . 6 ⊢ (R = S → (R ∘ R) = (S ∘ R))
3		coeq2 2503	. . . . . 6 ⊢ (R = S → (S ∘ R) = (S ∘ S))
4	2, 3	eqtrd 1128	. . . . 5 ⊢ (R = S → (R ∘ R) = (S ∘ S))
5	1, 4	uneq12d 1612	. . . 4 ⊢ (R = S → (◡R ∪ (R ∘ R)) = (◡S ∪ (S ∘ S)))
6	5	sseq1d 1527	. . 3 ⊢ (R = S → ((◡R ∪ (R ∘ R)) ⊆ R ↔ (◡S ∪ (S ∘ S)) ⊆ R))
7		sseq2 1522	. . 3 ⊢ (R = S → ((◡S ∪ (S ∘ S)) ⊆ R ↔ (◡S ∪ (S ∘ S)) ⊆ S))
8	6, 7	bitrd 406	. 2 ⊢ (R = S → ((◡R ∪ (R ∘ R)) ⊆ R ↔ (◡S ∪ (S ∘ S)) ⊆ S))
9		df-er 3200	. 2 ⊢ (Er R ↔ (◡R ∪ (R ∘ R)) ⊆ R)
10		df-er 3200	. 2 ⊢ (Er S ↔ (◡S ∪ (S ∘ S)) ⊆ S)
11	8, 9, 10	3bitr4g 428	1 ⊢ (R = S → (Er R ↔ Er S))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∪ cun 1485   ⊆ wss 1487  ^◡ccnv 2409   ∘ ccom 2414  Er wer 3197

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-co 2427  df-er 3200

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