Theorem resfval 17162

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 17131	. . 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 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → 𝑓 = 𝐹)
4	3	fveq2d 6674	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (1st ‘𝑓) = (1st ‘𝐹))
5		simprr 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ℎ = 𝐻)
6	5	dmeqd 5774	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → dom ℎ = dom 𝐻)
7	6	dmeqd 5774	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → dom dom ℎ = dom dom 𝐻)
8	4, 7	reseq12d 5854	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ((1st ‘𝑓) ↾ dom dom ℎ) = ((1st ‘𝐹) ↾ dom dom 𝐻))
9	3	fveq2d 6674	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (2nd ‘𝑓) = (2nd ‘𝐹))
10	9	fveq1d 6672	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → ((2nd ‘𝑓)‘𝑥) = ((2nd ‘𝐹)‘𝑥))
11	5	fveq1d 6672	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (ℎ‘𝑥) = (𝐻‘𝑥))
12	10, 11	reseq12d 5854	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ ℎ = 𝐻)) → (((2nd ‘𝑓)‘𝑥) ↾ (ℎ‘𝑥)) = (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))
13	6, 12	mpteq12dv 5151	. . 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 4811	. 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 3514	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
17		resfval.d	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
18	17	elexd 3514	. 2 ⊢ (𝜑 → 𝐻 ∈ V)
19		opex 5356	. . 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 7302	1 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑥) ↾ (𝐻‘𝑥)))⟩)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ⟨cop 4573   ↦ cmpt 5146  dom cdm 5555   ↾ cres 5557  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  1^st c1st 7687  2^nd c2nd 7688   ↾_f cresf 17127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-res 5567  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-resf 17131

This theorem is referenced by:  resfval2  17163  resf1st  17164  resf2nd  17165  funcres  17166