Theorem resfval 17854

Description: Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
resfval.c	⊢ (𝜑 → 𝐹 ∈ 𝑉)
resfval.d	⊢ (𝜑 → 𝐻 ∈ 𝑊)

Assertion

Ref	Expression
resfval	⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))⟩)

Distinct variable groups:   𝑥,𝐹   𝑥,𝐻   𝜑,𝑥

Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem resfval

Dummy variables 𝑓 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-resf 17823	. . 3 ⊢ ↾f = (𝑓 ∈ V, ℎ ∈ V ↦ ⟨((1st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩)
2	1	a1i 11	. 2 ⊢ (𝜑 → ↾f = (𝑓 ∈ V, ℎ ∈ V ↦ ⟨((1st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩))
3		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → 𝑓 = 𝐹)
4	3	fveq2d 6862	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (1st ‘𝑓) = (1st ‘𝐹))
5		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ℎ = 𝐻)
6	5	dmeqd 5869	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → dom ℎ = dom 𝐻)
7	6	dmeqd 5869	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → dom dom ℎ = dom dom 𝐻)
8	4, 7	reseq12d 5951	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ((1st ‘𝑓) ↾ dom dom ℎ) = ((1st ‘𝐹) ↾ dom dom 𝐻))
9	3	fveq2d 6862	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (2nd ‘𝑓) = (2nd ‘𝐹))
10	9	fveq1d 6860	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ((2nd ‘𝑓)‘𝑥) = ((2nd ‘𝐹)‘𝑥))
11	5	fveq1d 6860	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (ℎ‘𝑥) = (𝐻‘𝑥))
12	10, 11	reseq12d 5951	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)) = (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))
13	6, 12	mpteq12dv 5194	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥))) = (𝑥 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥))))
14	8, 13	opeq12d 4845	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ⟨((1st ‘𝑓) ↾ dom dom ℎ), (𝑥 ∈ dom ℎ ↦ (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)))⟩ = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))⟩)
15		resfval.c	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
16	15	elexd 3471	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
17		resfval.d	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
18	17	elexd 3471	. 2 ⊢ (𝜑 → 𝐻 ∈ V)
19		opex 5424	. . 3 ⊢ ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))⟩ ∈ V
20	19	a1i 11	. 2 ⊢ (𝜑 → ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))⟩ ∈ V)
21	2, 14, 16, 18, 20	ovmpod 7541	1 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   ↦ cmpt 5188  dom cdm 5638   ↾ cres 5640  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967   ↾_f cresf 17819

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-sep 5251  ax-nul 5261  ax-pr 5387

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-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-resf 17823

This theorem is referenced by:  resfval2  17855  resf1st  17856  resf2nd  17857  funcres  17858