Theorem resfval2 17657

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)
resfval2.g	⊢ (𝜑 → 𝐺 ∈ 𝑋)
resfval2.d	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))

Assertion

Ref	Expression
resfval2	⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹 ↾ 𝑆), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)

Distinct variable groups:   𝑥,𝐹   𝑥,𝑦,𝐺   𝑥,𝐻,𝑦   𝜑,𝑥   𝑥,𝑆,𝑦

Allowed substitution hints:   𝜑(𝑦)   𝐹(𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem resfval2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5392	. . . 4 ⊢ ⟨𝐹, 𝐺⟩ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
3		resfval.d	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
4	2, 3	resfval 17656	. 2 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
5		resfval.c	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
6		resfval2.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑋)
7		op1stg 7875	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
8	5, 6, 7	syl2anc 585	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
9		resfval2.d	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
10	9	fndmd 6569	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
11	10	dmeqd 5827	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12		dmxpid 5851	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
13	11, 12	eqtrdi 2792	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
14	8, 13	reseq12d 5904	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻) = (𝐹 ↾ 𝑆))
15		op2ndg 7876	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
16	5, 6, 15	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
17	16	fveq1d 6806	. . . . . 6 ⊢ (𝜑 → ((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) = (𝐺‘𝑧))
18	17	reseq1d 5902	. . . . 5 ⊢ (𝜑 → (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧)) = ((𝐺‘𝑧) ↾ (𝐻‘𝑧)))
19	10, 18	mpteq12dv 5172	. . . 4 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺‘𝑧) ↾ (𝐻‘𝑧))))
20		fveq2 6804	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
21		df-ov 7310	. . . . . . 7 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
22	20, 21	eqtr4di 2794	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
23		fveq2 6804	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
24		df-ov 7310	. . . . . . 7 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
25	23, 24	eqtr4di 2794	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
26	22, 25	reseq12d 5904	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ↾ (𝐻‘𝑧)) = ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
27	26	mpompt 7420	. . . 4 ⊢ (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺‘𝑧) ↾ (𝐻‘𝑧))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
28	19, 27	eqtrdi 2792	. . 3 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦))))
29	14, 28	opeq12d 4817	. 2 ⊢ (𝜑 → ⟨((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧)))⟩ = ⟨(𝐹 ↾ 𝑆), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
30	4, 29	eqtrd 2776	1 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹 ↾ 𝑆), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  Vcvv 3437  ⟨cop 4571   ↦ cmpt 5164   × cxp 5598  dom cdm 5600   ↾ cres 5602   Fn wfn 6453  ‘cfv 6458  (class class class)co 7307   ∈ cmpo 7309  1^st c1st 7861  2^nd c2nd 7862   ↾_f cresf 17621

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-iota 6410  df-fun 6460  df-fn 6461  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-1st 7863  df-2nd 7864  df-resf 17625

This theorem is referenced by:  funcrngcsetc  45800  funcringcsetc  45837