Theorem coi2 5187

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		cnvco 4852	. . 3 ⊢ ◡(◡𝐴 ∘ I ) = (◡ I ∘ ◡◡𝐴)
2		relcnv 5048	. . . . 5 ⊢ Rel ◡𝐴
3		coi1 5186	. . . . 5 ⊢ (Rel ◡𝐴 → (◡𝐴 ∘ I ) = ◡𝐴)
4	2, 3	ax-mp 5	. . . 4 ⊢ (◡𝐴 ∘ I ) = ◡𝐴
5	4	cnveqi 4842	. . 3 ⊢ ◡(◡𝐴 ∘ I ) = ◡◡𝐴
6	1, 5	eqtr3i 2219	. 2 ⊢ (◡ I ∘ ◡◡𝐴) = ◡◡𝐴
7		dfrel2 5121	. . 3 ⊢ (Rel 𝐴 ↔ ◡◡𝐴 = 𝐴)
8		cnvi 5075	. . . 4 ⊢ ◡ I = I
9		coeq2 4825	. . . . 5 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = (◡ I ∘ 𝐴))
10		coeq1 4824	. . . . 5 ⊢ (◡ I = I → (◡ I ∘ 𝐴) = ( I ∘ 𝐴))
11	9, 10	sylan9eq 2249	. . . 4 ⊢ ((◡◡𝐴 = 𝐴 ∧ ◡ I = I ) → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
12	8, 11	mpan2 425	. . 3 ⊢ (◡◡𝐴 = 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
13	7, 12	sylbi 121	. 2 ⊢ (Rel 𝐴 → (◡ I ∘ ◡◡𝐴) = ( I ∘ 𝐴))
14	7	biimpi 120	. 2 ⊢ (Rel 𝐴 → ◡◡𝐴 = 𝐴)
15	6, 13, 14	3eqtr3a 2253	1 ⊢ (Rel 𝐴 → ( I ∘ 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1364   I cid 4324  ^◡ccnv 4663   ∘ ccom 4668  Rel wrel 4669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673

This theorem is referenced by:  relcoi2  5201  funi  5291  fcoi2  5442