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Theorem resfval2 17850
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 5464 . . . 4 𝐹, 𝐺⟩ ∈ V
21a1i 11 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
3 resfval.d . . 3 (𝜑𝐻𝑊)
42, 3resfval 17849 . 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 7991 . . . . 5 ((𝐹𝑉𝐺𝑋) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
85, 6, 7syl2anc 583 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
9 resfval2.d . . . . . . 7 (𝜑𝐻 Fn (𝑆 × 𝑆))
109fndmd 6654 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1110dmeqd 5905 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12 dmxpid 5929 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1311, 12eqtrdi 2787 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
148, 13reseq12d 5982 . . 3 (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻) = (𝐹𝑆))
15 op2ndg 7992 . . . . . . . 8 ((𝐹𝑉𝐺𝑋) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
165, 6, 15syl2anc 583 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
1716fveq1d 6893 . . . . . 6 (𝜑 → ((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) = (𝐺𝑧))
1817reseq1d 5980 . . . . 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 6891 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
21 df-ov 7415 . . . . . . 7 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
2220, 21eqtr4di 2789 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
23 fveq2 6891 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
24 df-ov 7415 . . . . . . 7 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
2523, 24eqtr4di 2789 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
2622, 25reseq12d 5982 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ↾ (𝐻𝑧)) = ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2726mpompt 7525 . . . 4 (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺𝑧) ↾ (𝐻𝑧))) = (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2819, 27eqtrdi 2787 . . 3 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧))) = (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦))))
2914, 28opeq12d 4881 . 2 (𝜑 → ⟨((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧)))⟩ = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
304, 29eqtrd 2771 1 (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  Vcvv 3473  cop 4634  cmpt 5231   × cxp 5674  dom cdm 5676  cres 5678   Fn wfn 6538  cfv 6543  (class class class)co 7412  cmpo 7414  1st c1st 7977  2nd c2nd 7978  f cresf 17814
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-resf 17818
This theorem is referenced by:  funcrngcsetc  20532  funcringcsetc  20566
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