Theorem resfval2 17589

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 5381	. . . 4 ⊢ ⟨𝐹, 𝐺⟩ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
3		resfval.d	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
4	2, 3	resfval 17588	. 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 7829	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
8	5, 6, 7	syl2anc 583	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
9		resfval2.d	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
10	9	fndmd 6534	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
11	10	dmeqd 5811	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12		dmxpid 5836	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
13	11, 12	eqtrdi 2795	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
14	8, 13	reseq12d 5889	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻) = (𝐹 ↾ 𝑆))
15		op2ndg 7830	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
16	5, 6, 15	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
17	16	fveq1d 6770	. . . . . 6 ⊢ (𝜑 → ((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) = (𝐺‘𝑧))
18	17	reseq1d 5887	. . . . 5 ⊢ (𝜑 → (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧)) = ((𝐺‘𝑧) ↾ (𝐻‘𝑧)))
19	10, 18	mpteq12dv 5169	. . . 4 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺‘𝑧) ↾ (𝐻‘𝑧))))
20		fveq2 6768	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
21		df-ov 7271	. . . . . . 7 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
22	20, 21	eqtr4di 2797	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
23		fveq2 6768	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
24		df-ov 7271	. . . . . . 7 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
25	23, 24	eqtr4di 2797	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
26	22, 25	reseq12d 5889	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ↾ (𝐻‘𝑧)) = ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
27	26	mpompt 7379	. . . 4 ⊢ (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺‘𝑧) ↾ (𝐻‘𝑧))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
28	19, 27	eqtrdi 2795	. . 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 2779	1 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹 ↾ 𝑆), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2109  Vcvv 3430  ⟨cop 4572   ↦ cmpt 5161   × cxp 5586  dom cdm 5588   ↾ cres 5590   Fn wfn 6425  ‘cfv 6430  (class class class)co 7268   ∈ cmpo 7270  1^st c1st 7815  2^nd c2nd 7816   ↾_f cresf 17553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-iota 6388  df-fun 6432  df-fn 6433  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-resf 17557

This theorem is referenced by:  funcrngcsetc  45508  funcringcsetc  45545