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Theorem resfval2 17888
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 5470 . . . 4 𝐹, 𝐺⟩ ∈ V
21a1i 11 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
3 resfval.d . . 3 (𝜑𝐻𝑊)
42, 3resfval 17887 . 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 8013 . . . . 5 ((𝐹𝑉𝐺𝑋) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
85, 6, 7syl2anc 582 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
9 resfval2.d . . . . . . 7 (𝜑𝐻 Fn (𝑆 × 𝑆))
109fndmd 6664 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1110dmeqd 5912 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12 dmxpid 5936 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1311, 12eqtrdi 2784 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
148, 13reseq12d 5990 . . 3 (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻) = (𝐹𝑆))
15 op2ndg 8014 . . . . . . . 8 ((𝐹𝑉𝐺𝑋) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
165, 6, 15syl2anc 582 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
1716fveq1d 6904 . . . . . 6 (𝜑 → ((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) = (𝐺𝑧))
1817reseq1d 5988 . . . . 5 (𝜑 → (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧)) = ((𝐺𝑧) ↾ (𝐻𝑧)))
1910, 18mpteq12dv 5243 . . . 4 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺𝑧) ↾ (𝐻𝑧))))
20 fveq2 6902 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
21 df-ov 7429 . . . . . . 7 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
2220, 21eqtr4di 2786 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
23 fveq2 6902 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
24 df-ov 7429 . . . . . . 7 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
2523, 24eqtr4di 2786 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
2622, 25reseq12d 5990 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ↾ (𝐻𝑧)) = ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2726mpompt 7541 . . . 4 (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺𝑧) ↾ (𝐻𝑧))) = (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2819, 27eqtrdi 2784 . . 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 2768 1 (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3473  cop 4638  cmpt 5235   × cxp 5680  dom cdm 5682  cres 5684   Fn wfn 6548  cfv 6553  (class class class)co 7426  cmpo 7428  1st c1st 7999  2nd c2nd 8000  f cresf 17852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-resf 17856
This theorem is referenced by:  funcrngcsetc  20587  funcringcsetc  20621
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