Theorem resfval2 17946

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 5443	. . . 4 ⊢ ⟨𝐹, 𝐺⟩ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
3		resfval.d	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
4	2, 3	resfval 17945	. 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 7994	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
8	5, 6, 7	syl2anc 595	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
9		resfval2.d	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
10	9	fndmd 6638	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
11	10	dmeqd 5893	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12		dmxpid 5918	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
13	11, 12	eqtrdi 2820	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
14	8, 13	reseq12d 5977	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻) = (𝐹 ↾ 𝑆))
15		op2ndg 7995	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
16	5, 6, 15	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
17	16	fveq1d 6881	. . . . . 6 ⊢ (𝜑 → ((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) = (𝐺‘𝑧))
18	17	reseq1d 5975	. . . . 5 ⊢ (𝜑 → (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧)) = ((𝐺‘𝑧) ↾ (𝐻‘𝑧)))
19	10, 18	mpteq12dv 5199	. . . 4 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺‘𝑧) ↾ (𝐻‘𝑧))))
20		fveq2 6879	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
21		df-ov 7411	. . . . . . 7 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
22	20, 21	eqtr4di 2822	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
23		fveq2 6879	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
24		df-ov 7411	. . . . . . 7 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
25	23, 24	eqtr4di 2822	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
26	22, 25	reseq12d 5977	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ↾ (𝐻‘𝑧)) = ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
27	26	mpompt 7522	. . . 4 ⊢ (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺‘𝑧) ↾ (𝐻‘𝑧))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
28	19, 27	eqtrdi 2820	. . 3 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦))))
29	14, 28	opeq12d 4847	. 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 2804	1 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹 ↾ 𝑆), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4597   ↦ cmpt 5193   × cxp 5657  dom cdm 5659   ↾ cres 5661   Fn wfn 6528  ‘cfv 6533  (class class class)co 7408   ∈ cmpo 7410  1^st c1st 7980  2^nd c2nd 7981   ↾_f cresf 17910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-resf 17914

This theorem is referenced by:  funcrngcsetc  20721  funcringcsetc  20755