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Theorem resfval2 17944
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 5475 . . . 4 𝐹, 𝐺⟩ ∈ V
21a1i 11 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
3 resfval.d . . 3 (𝜑𝐻𝑊)
42, 3resfval 17943 . 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 8025 . . . . 5 ((𝐹𝑉𝐺𝑋) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
85, 6, 7syl2anc 584 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
9 resfval2.d . . . . . . 7 (𝜑𝐻 Fn (𝑆 × 𝑆))
109fndmd 6674 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1110dmeqd 5919 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12 dmxpid 5944 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1311, 12eqtrdi 2791 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
148, 13reseq12d 6001 . . 3 (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻) = (𝐹𝑆))
15 op2ndg 8026 . . . . . . . 8 ((𝐹𝑉𝐺𝑋) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
165, 6, 15syl2anc 584 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
1716fveq1d 6909 . . . . . 6 (𝜑 → ((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) = (𝐺𝑧))
1817reseq1d 5999 . . . . 5 (𝜑 → (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧)) = ((𝐺𝑧) ↾ (𝐻𝑧)))
1910, 18mpteq12dv 5239 . . . 4 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺𝑧) ↾ (𝐻𝑧))))
20 fveq2 6907 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
21 df-ov 7434 . . . . . . 7 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
2220, 21eqtr4di 2793 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
23 fveq2 6907 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
24 df-ov 7434 . . . . . . 7 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
2523, 24eqtr4di 2793 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
2622, 25reseq12d 6001 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ↾ (𝐻𝑧)) = ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2726mpompt 7547 . . . 4 (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺𝑧) ↾ (𝐻𝑧))) = (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2819, 27eqtrdi 2791 . . 3 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧))) = (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦))))
2914, 28opeq12d 4886 . 2 (𝜑 → ⟨((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧)))⟩ = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
304, 29eqtrd 2775 1 (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cmpt 5231   × cxp 5687  dom cdm 5689  cres 5691   Fn wfn 6558  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  f cresf 17908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-resf 17912
This theorem is referenced by:  funcrngcsetc  20657  funcringcsetc  20691
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