Theorem idafval 18026

Description: Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
idafval.i	⊢ 𝐼 = (Ida‘𝐶)
idafval.b	⊢ 𝐵 = (Base‘𝐶)
idafval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
idafval.1	⊢ 1 = (Id‘𝐶)

Assertion

Ref	Expression
idafval	⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))

Distinct variable groups:   𝑥, 1   𝑥,𝐵   𝑥,𝐶   𝑥,𝐼   𝜑,𝑥

Proof of Theorem idafval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		idafval.i	. 2 ⊢ 𝐼 = (Ida‘𝐶)
2		idafval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		fveq2 6861	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		idafval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2783	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fveq2 6861	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶))
7		idafval.1	. . . . . . . 8 ⊢ 1 = (Id‘𝐶)
8	6, 7	eqtr4di 2783	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = 1 )
9	8	fveq1d 6863	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥))
10	9	oteq3d 4854	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩ = ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩)
11	5, 10	mpteq12dv 5197	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩) = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
12		df-ida 18024	. . . 4 ⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
13	11, 12, 4	mptfvmpt 7205	. . 3 ⊢ (𝐶 ∈ Cat → (Ida‘𝐶) = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
14	2, 13	syl 17	. 2 ⊢ (𝜑 → (Ida‘𝐶) = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
15	1, 14	eqtrid 2777	1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cotp 4600   ↦ cmpt 5191  ‘cfv 6514  Basecbs 17186  Catccat 17632  Idccid 17633  Id_acida 18022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-ot 4601  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ida 18024

This theorem is referenced by:  idaval  18027  idaf  18032