Theorem coi2 6195

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 6121	. 2 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
2		cnvco 5821	. . . 4 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴)
3		relcnv 6036	. . . . . 6 ⊢ Rel ◡𝐴
4		coi1 6194	. . . . . 6 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴)
5	3, 4	ax-mp 5	. . . . 5 ⊢ (◡𝐴 ∘ I ) = ◡𝐴
6	5	cnveqi 5810	. . . 4 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴
7	2, 6	eqtr3i 2766	. . 3 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴
8		cnvi 6074	. . . 4 ⊢ ◡ I = I
9		coeq2 5794	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴))
10		coeq1 5793	. . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴))
11	9, 10	sylan9eq 2796	. . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
12	8, 11	mpan2 688	. . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
13		id 22	. . 3 ⊢ (◡◡𝐴 = 𝐴 → ◡◡𝐴 = 𝐴)
14	7, 12, 13	3eqtr3a 2800	. 2 ⊢ (◡◡𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴)
15	1, 14	sylbi 216	1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   I cid 5511  ^◡ccnv 5613   ∘ ccom 5618  Rel wrel 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-br 5090  df-opab 5152  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623

This theorem is referenced by:  relcoi2  6209  funi  6510  fcoi2  6694