Theorem resfval 17523

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 17492	. . 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 767	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → 𝑓 = 𝐹)
4	3	fveq2d 6760	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (1st ‘𝑓) = (1st ‘𝐹))
5		simprr 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ℎ = 𝐻)
6	5	dmeqd 5803	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → dom ℎ = dom 𝐻)
7	6	dmeqd 5803	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → dom dom ℎ = dom dom 𝐻)
8	4, 7	reseq12d 5881	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ((1st ‘𝑓) ↾ dom dom ℎ) = ((1st ‘𝐹) ↾ dom dom 𝐻))
9	3	fveq2d 6760	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (2nd ‘𝑓) = (2nd ‘𝐹))
10	9	fveq1d 6758	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ((2nd ‘𝑓)‘𝑥) = ((2nd ‘𝐹)‘𝑥))
11	5	fveq1d 6758	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (ℎ‘𝑥) = (𝐻‘𝑥))
12	10, 11	reseq12d 5881	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)) = (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))
13	6, 12	mpteq12dv 5161	. . 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 4809	. 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 3442	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
17		resfval.d	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
18	17	elexd 3442	. 2 ⊢ (𝜑 → 𝐻 ∈ V)
19		opex 5373	. . 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 7403	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 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   ↦ cmpt 5153  dom cdm 5580   ↾ cres 5582  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1^st c1st 7802  2^nd c2nd 7803   ↾_f cresf 17488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-resf 17492

This theorem is referenced by:  resfval2  17524  resf1st  17525  resf2nd  17526  funcres  17527