Theorem idafval 17982

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 6826	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		idafval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2782	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fveq2 6826	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶))
7		idafval.1	. . . . . . . 8 ⊢ 1 = (Id‘𝐶)
8	6, 7	eqtr4di 2782	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = 1 )
9	8	fveq1d 6828	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((Id‘𝑐)‘𝑥) = ( 1 ‘𝑥))
10	9	oteq3d 4841	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩ = ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩)
11	5, 10	mpteq12dv 5182	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩) = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
12		df-ida 17980	. . . 4 ⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
13	11, 12, 4	mptfvmpt 7168	. . 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 2776	1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cotp 4587   ↦ cmpt 5176  ‘cfv 6486  Basecbs 17138  Catccat 17588  Idccid 17589  Id_acida 17978

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 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ida 17980

This theorem is referenced by:  idaval  17983  idaf  17988