Theorem resfval 17943

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 17912	. . 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 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → 𝑓 = 𝐹)
4	3	fveq2d 6911	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (1st ‘𝑓) = (1st ‘𝐹))
5		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ℎ = 𝐻)
6	5	dmeqd 5919	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → dom ℎ = dom 𝐻)
7	6	dmeqd 5919	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → dom dom ℎ = dom dom 𝐻)
8	4, 7	reseq12d 6001	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ((1st ‘𝑓) ↾ dom dom ℎ) = ((1st ‘𝐹) ↾ dom dom 𝐻))
9	3	fveq2d 6911	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (2nd ‘𝑓) = (2nd ‘𝐹))
10	9	fveq1d 6909	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ((2nd ‘𝑓)‘𝑥) = ((2nd ‘𝐹)‘𝑥))
11	5	fveq1d 6909	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (ℎ‘𝑥) = (𝐻‘𝑥))
12	10, 11	reseq12d 6001	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)) = (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))
13	6, 12	mpteq12dv 5239	. . 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 4886	. 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 3502	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
17		resfval.d	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
18	17	elexd 3502	. 2 ⊢ (𝜑 → 𝐻 ∈ V)
19		opex 5475	. . 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 7585	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 1537   ∈ wcel 2106  Vcvv 3478  ⟨cop 4637   ↦ cmpt 5231  dom cdm 5689   ↾ cres 5691  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  1^st c1st 8011  2^nd c2nd 8012   ↾_f cresf 17908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-resf 17912

This theorem is referenced by:  resfval2  17944  resf1st  17945  resf2nd  17946  funcres  17947