Theorem idafval 18136

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 6885	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4		idafval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	eqtr4di 2818	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6		fveq2 6885	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = (Id‘𝐶))
7		idafval.1	. . . . . . . 8 ⊢ 1 = (Id‘𝐶)
8	6, 7	eqtr4di 2818	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Id‘𝑐) = 1 )
9	8	fveq1d 6887	. . . . . 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 5200	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩) = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
12		df-ida 18134	. . . 4 ⊢ Ida = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐) ↦ ⟨𝑥, 𝑥, ((Id‘𝑐)‘𝑥)⟩))
13	11, 12, 4	mptfvmpt 7233	. . 3 ⊢ (𝐶 ∈ Cat → (Ida‘𝐶) = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
14	2, 13	syl 18	. 2 ⊢ (𝜑 → (Ida‘𝐶) = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))
15	1, 14	eqtrid 2812	1 ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ ⟨𝑥, 𝑥, ( 1 ‘𝑥)⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ⟨cotp 4599   ↦ cmpt 5194  ‘cfv 6540  Basecbs 17291  Catccat 17742  Idccid 17743  Id_acida 18132

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-ot 4600  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ida 18134

This theorem is used by:  idaval  18137  idaf  18142