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Theorem resfval2 17237
 Description: Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resfval.c (𝜑𝐹𝑉)
resfval.d (𝜑𝐻𝑊)
resfval2.g (𝜑𝐺𝑋)
resfval2.d (𝜑𝐻 Fn (𝑆 × 𝑆))
Assertion
Ref Expression
resfval2 (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((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.
StepHypRef Expression
1 opex 5329 . . . 4 𝐹, 𝐺⟩ ∈ V
21a1i 11 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
3 resfval.d . . 3 (𝜑𝐻𝑊)
42, 3resfval 17236 . 2 (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧)))⟩)
5 resfval.c . . . . 5 (𝜑𝐹𝑉)
6 resfval2.g . . . . 5 (𝜑𝐺𝑋)
7 op1stg 7712 . . . . 5 ((𝐹𝑉𝐺𝑋) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
85, 6, 7syl2anc 587 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
9 resfval2.d . . . . . . 7 (𝜑𝐻 Fn (𝑆 × 𝑆))
109fndmd 6444 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1110dmeqd 5752 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12 dmxpid 5777 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1311, 12eqtrdi 2810 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
148, 13reseq12d 5830 . . 3 (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻) = (𝐹𝑆))
15 op2ndg 7713 . . . . . . . 8 ((𝐹𝑉𝐺𝑋) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
165, 6, 15syl2anc 587 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
1716fveq1d 6666 . . . . . 6 (𝜑 → ((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) = (𝐺𝑧))
1817reseq1d 5828 . . . . 5 (𝜑 → (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧)) = ((𝐺𝑧) ↾ (𝐻𝑧)))
1910, 18mpteq12dv 5122 . . . 4 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺𝑧) ↾ (𝐻𝑧))))
20 fveq2 6664 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
21 df-ov 7160 . . . . . . 7 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
2220, 21eqtr4di 2812 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
23 fveq2 6664 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
24 df-ov 7160 . . . . . . 7 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
2523, 24eqtr4di 2812 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
2622, 25reseq12d 5830 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ↾ (𝐻𝑧)) = ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2726mpompt 7267 . . . 4 (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺𝑧) ↾ (𝐻𝑧))) = (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2819, 27eqtrdi 2810 . . 3 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧))) = (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦))))
2914, 28opeq12d 4775 . 2 (𝜑 → ⟨((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧)))⟩ = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
304, 29eqtrd 2794 1 (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2112  Vcvv 3410  ⟨cop 4532   ↦ cmpt 5117   × cxp 5527  dom cdm 5529   ↾ cres 5531   Fn wfn 6336  ‘cfv 6341  (class class class)co 7157   ∈ cmpo 7159  1st c1st 7698  2nd c2nd 7699   ↾f cresf 17201 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303  ax-un 7466 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-iun 4889  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-iota 6300  df-fun 6343  df-fn 6344  df-fv 6349  df-ov 7160  df-oprab 7161  df-mpo 7162  df-1st 7700  df-2nd 7701  df-resf 17205 This theorem is referenced by:  funcrngcsetc  45049  funcringcsetc  45086
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