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Theorem resfval 17841
Description: Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resfval.c (𝜑𝐹𝑉)
resfval.d (𝜑𝐻𝑊)
Assertion
Ref Expression
resfval (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))⟩)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐻   𝜑,𝑥
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem resfval
Dummy variables 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-resf 17810 . . 3 f = (𝑓 ∈ V, ∈ V ↦ ⟨((1st𝑓) ↾ dom dom ), (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥)))⟩)
21a1i 11 . 2 (𝜑 → ↾f = (𝑓 ∈ V, ∈ V ↦ ⟨((1st𝑓) ↾ dom dom ), (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥)))⟩))
3 simprl 769 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → 𝑓 = 𝐹)
43fveq2d 6895 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (1st𝑓) = (1st𝐹))
5 simprr 771 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → = 𝐻)
65dmeqd 5905 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom = dom 𝐻)
76dmeqd 5905 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom dom = dom dom 𝐻)
84, 7reseq12d 5982 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((1st𝑓) ↾ dom dom ) = ((1st𝐹) ↾ dom dom 𝐻))
93fveq2d 6895 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (2nd𝑓) = (2nd𝐹))
109fveq1d 6893 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((2nd𝑓)‘𝑥) = ((2nd𝐹)‘𝑥))
115fveq1d 6893 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥) = (𝐻𝑥))
1210, 11reseq12d 5982 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (((2nd𝑓)‘𝑥) ↾ (𝑥)) = (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))
136, 12mpteq12dv 5239 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥))) = (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥))))
148, 13opeq12d 4881 . 2 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ⟨((1st𝑓) ↾ dom dom ), (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥)))⟩ = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))⟩)
15 resfval.c . . 3 (𝜑𝐹𝑉)
1615elexd 3494 . 2 (𝜑𝐹 ∈ V)
17 resfval.d . . 3 (𝜑𝐻𝑊)
1817elexd 3494 . 2 (𝜑𝐻 ∈ V)
19 opex 5464 . . 3 ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))⟩ ∈ V
2019a1i 11 . 2 (𝜑 → ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))⟩ ∈ V)
212, 14, 16, 18, 20ovmpod 7559 1 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cop 4634  cmpt 5231  dom cdm 5676  cres 5678  cfv 6543  (class class class)co 7408  cmpo 7410  1st c1st 7972  2nd c2nd 7973  f cresf 17806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  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-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-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-resf 17810
This theorem is referenced by:  resfval2  17842  resf1st  17843  resf2nd  17844  funcres  17845
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