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Theorem resfval2 16760
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 5060 . . . 4 𝐹, 𝐺⟩ ∈ V
21a1i 11 . . 3 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ V)
3 resfval.d . . 3 (𝜑𝐻𝑊)
42, 3resfval 16759 . 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 7327 . . . . 5 ((𝐹𝑉𝐺𝑋) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
85, 6, 7syl2anc 573 . . . 4 (𝜑 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
9 resfval2.d . . . . . . 7 (𝜑𝐻 Fn (𝑆 × 𝑆))
10 fndm 6130 . . . . . . 7 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
119, 10syl 17 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1211dmeqd 5464 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
13 dmxpid 5483 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1412, 13syl6eq 2821 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
158, 14reseq12d 5535 . . 3 (𝜑 → ((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻) = (𝐹𝑆))
16 op2ndg 7328 . . . . . . . 8 ((𝐹𝑉𝐺𝑋) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
175, 6, 16syl2anc 573 . . . . . . 7 (𝜑 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
1817fveq1d 6334 . . . . . 6 (𝜑 → ((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) = (𝐺𝑧))
1918reseq1d 5533 . . . . 5 (𝜑 → (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧)) = ((𝐺𝑧) ↾ (𝐻𝑧)))
2011, 19mpteq12dv 4867 . . . 4 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧))) = (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺𝑧) ↾ (𝐻𝑧))))
21 fveq2 6332 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝐺‘⟨𝑥, 𝑦⟩))
22 df-ov 6796 . . . . . . 7 (𝑥𝐺𝑦) = (𝐺‘⟨𝑥, 𝑦⟩)
2321, 22syl6eqr 2823 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐺𝑧) = (𝑥𝐺𝑦))
24 fveq2 6332 . . . . . . 7 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝐻‘⟨𝑥, 𝑦⟩))
25 df-ov 6796 . . . . . . 7 (𝑥𝐻𝑦) = (𝐻‘⟨𝑥, 𝑦⟩)
2624, 25syl6eqr 2823 . . . . . 6 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐻𝑧) = (𝑥𝐻𝑦))
2723, 26reseq12d 5535 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝐺𝑧) ↾ (𝐻𝑧)) = ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2827mpt2mpt 6899 . . . 4 (𝑧 ∈ (𝑆 × 𝑆) ↦ ((𝐺𝑧) ↾ (𝐻𝑧))) = (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))
2920, 28syl6eq 2821 . . 3 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧))) = (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦))))
3015, 29opeq12d 4547 . 2 (𝜑 → ⟨((1st ‘⟨𝐹, 𝐺⟩) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘⟨𝐹, 𝐺⟩)‘𝑧) ↾ (𝐻𝑧)))⟩ = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
314, 30eqtrd 2805 1 (𝜑 → (⟨𝐹, 𝐺⟩ ↾f 𝐻) = ⟨(𝐹𝑆), (𝑥𝑆, 𝑦𝑆 ↦ ((𝑥𝐺𝑦) ↾ (𝑥𝐻𝑦)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322  cmpt 4863   × cxp 5247  dom cdm 5249  cres 5251   Fn wfn 6026  cfv 6031  (class class class)co 6793  cmpt2 6795  1st c1st 7313  2nd c2nd 7314  f cresf 16724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-resf 16728
This theorem is referenced by:  funcrngcsetc  42526  funcringcsetc  42563
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