Theorem coi2 2666

Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36.

Assertion

Ref	Expression
coi2	⊢ (Rel A → (I ∘ A) = A)

Proof of Theorem coi2

Step	Hyp	Ref	Expression
1		dfrel2 2660	. . . 4 ⊢ (Rel A ↔ ◡◡A = A)
2		cnvi 2634	. . . . 5 ⊢ ◡I = I
3		coeq2 2503	. . . . . 6 ⊢ (◡◡A = A → (◡I ∘ ◡◡A) = (◡I ∘ A))
4		coeq1 2502	. . . . . 6 ⊢ (◡I = I → (◡I ∘ A) = (I ∘ A))
5	3, 4	sylan9eq 1144	. . . . 5 ⊢ ((◡◡A = A ∧ ◡I = I) → (◡I ∘ ◡◡A) = (I ∘ A))
6	2, 5	mpan2 519	. . . 4 ⊢ (◡◡A = A → (◡I ∘ ◡◡A) = (I ∘ A))
7	1, 6	sylbi 174	. . 3 ⊢ (Rel A → (◡I ∘ ◡◡A) = (I ∘ A))
8		cnvco 2520	. . . 4 ⊢ ◡(◡A ∘ I) = (◡I ∘ ◡◡A)
9		relcnv 2624	. . . . . 6 ⊢ Rel ◡A
10		coi1 2665	. . . . . 6 ⊢ (Rel ◡A → (◡A ∘ I) = ◡A)
11	9, 10	ax-mp 6	. . . . 5 ⊢ (◡A ∘ I) = ◡A
12		cnveq 2513	. . . . 5 ⊢ ((◡A ∘ I) = ◡A → ◡(◡A ∘ I) = ◡◡A)
13	11, 12	ax-mp 6	. . . 4 ⊢ ◡(◡A ∘ I) = ◡◡A
14	8, 13	eqtr3 1121	. . 3 ⊢ (◡I ∘ ◡◡A) = ◡◡A
15	7, 14	syl5reqr 1139	. 2 ⊢ (Rel A → (I ∘ A) = ◡◡A)
16	1	biimp 133	. 2 ⊢ (Rel A → ◡◡A = A)
17	15, 16	eqtrd 1128	1 ⊢ (Rel A → (I ∘ A) = A)

Syntax hints:   → wi 2   = wceq 1091  Icid 2057  ^◡ccnv 2409   ∘ ccom 2414  Rel wrel 2415

This theorem is referenced by:  funi 2692

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-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427

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