Theorem coi2 6254

Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)

Assertion

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

Proof of Theorem coi2

Step	Hyp	Ref	Expression
1		dfrel2 6178	. 2 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
2		cnvco 5865	. . . 4 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴)
3		relcnv 6096	. . . . . 6 ⊢ Rel ◡𝐴
4		coi1 6253	. . . . . 6 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴)
5	3, 4	ax-mp 5	. . . . 5 ⊢ (◡𝐴 ∘ I ) = ◡𝐴
6	5	cnveqi 5850	. . . 4 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴
7	2, 6	eqtr3i 2790	. . 3 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴
8		cnvi 5861	. . . 4 ⊢ ◡ I = I
9		coeq2 5834	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴))
10		coeq1 5833	. . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴))
11	9, 10	sylan9eq 2820	. . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
12	8, 11	mpan2 703	. . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
13		id 23	. . 3 ⊢ (◡◡𝐴 = 𝐴 → ◡◡𝐴 = 𝐴)
14	7, 12, 13	3eqtr3a 2824	. 2 ⊢ (◡◡𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴)
15	1, 14	sylbi 220	1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1563   I cid 5545  ^◡ccnv 5650   ∘ ccom 5655  Rel wrel 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660

This theorem is referenced by:  relcoi2  6267  funi  6557  fcoi2  6743