Theorem resfval2 17163

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 5356	. . . 4 ⊢ ⟨𝐹, 𝐺⟩ ∈ V
2	1	a1i 11	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
3		resfval.d	. . 3 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
4	2, 3	resfval 17162	. 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 7701	. . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
8	5, 6, 7	syl2anc 586	. . . 4 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
9		resfval2.d	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
10		fndm 6455	. . . . . . 7 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
11	9, 10	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
12	11	dmeqd 5774	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
13		dmxpid 5800	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
14	12, 13	syl6eq 2872	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
15	8, 14	reseq12d 5854	. . 3 ⊢ (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻) = (𝐹 ↾ 𝑆))
16		op2ndg 7702	. . . . . . . 8 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
17	5, 6, 16	syl2anc 586	. . . . . . 7 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
18	17	fveq1d 6672	. . . . . 6 ⊢ (𝜑 → ((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) = (𝐺‘𝑧))
19	18	reseq1d 5852	. . . . 5 ⊢ (𝜑 → (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧)) = ((𝐺‘𝑧) ↾ (𝐻‘𝑧)))
20	11, 19	mpteq12dv 5151	. . . 4 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺‘𝑧) ↾ (𝐻‘𝑧))))
21		fveq2 6670	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
22		df-ov 7159	. . . . . . 7 ⊢ (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
23	21, 22	syl6eqr 2874	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺‘𝑧) = (𝑥𝐺𝑦))
24		fveq2 6670	. . . . . . 7 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
25		df-ov 7159	. . . . . . 7 ⊢ (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
26	24, 25	syl6eqr 2874	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻‘𝑧) = (𝑥𝐻𝑦))
27	23, 26	reseq12d 5854	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺‘𝑧) ↾ (𝐻‘𝑧)) = ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
28	27	mpompt 7266	. . . 4 ⊢ (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺‘𝑧) ↾ (𝐻‘𝑧))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
29	20, 28	syl6eq 2872	. . 3 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻‘𝑧))) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦))))
30	15, 29	opeq12d 4811	. 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)))⟩)
31	4, 30	eqtrd 2856	1 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹 ↾ 𝑆), (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ⟨cop 4573   ↦ cmpt 5146   × cxp 5553  dom cdm 5555   ↾ cres 5557   Fn wfn 6350  ‘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-pow 5266  ax-pr 5330  ax-un 7461

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-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  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-iun 4921  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-rn 5566  df-res 5567  df-iota 6314  df-fun 6357  df-fn 6358  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-resf 17131

This theorem is referenced by:  funcrngcsetc  44289  funcringcsetc  44326