Theorem coi2 6118

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 6048	. 2 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
2		cnvco 5758	. . . 4 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴)
3		relcnv 5969	. . . . . 6 ⊢ Rel ◡𝐴
4		coi1 6117	. . . . . 6 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴)
5	3, 4	ax-mp 5	. . . . 5 ⊢ (◡𝐴 ∘ I ) = ◡𝐴
6	5	cnveqi 5747	. . . 4 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴
7	2, 6	eqtr3i 2848	. . 3 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴
8		cnvi 6002	. . . 4 ⊢ ◡ I = I
9		coeq2 5731	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴))
10		coeq1 5730	. . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴))
11	9, 10	sylan9eq 2878	. . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
12	8, 11	mpan2 689	. . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
13		id 22	. . 3 ⊢ (◡◡𝐴 = 𝐴 → ◡◡𝐴 = 𝐴)
14	7, 12, 13	3eqtr3a 2882	. 2 ⊢ (◡◡𝐴 = 𝐴 → ( I ∘ 𝐴) = 𝐴)
15	1, 14	sylbi 219	1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   I cid 5461  ^◡ccnv 5556   ∘ ccom 5561  Rel wrel 5562

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566

This theorem is referenced by:  relcoi2  6130  funi  6389  fcoi2  6555