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Theorem resfval 16758
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 16727 . . 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 746 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → 𝑓 = 𝐹)
43fveq2d 6336 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (1st𝑓) = (1st𝐹))
5 simprr 748 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → = 𝐻)
65dmeqd 5464 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom = dom 𝐻)
76dmeqd 5464 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom dom = dom dom 𝐻)
84, 7reseq12d 5535 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((1st𝑓) ↾ dom dom ) = ((1st𝐹) ↾ dom dom 𝐻))
93fveq2d 6336 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (2nd𝑓) = (2nd𝐹))
109fveq1d 6334 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((2nd𝑓)‘𝑥) = ((2nd𝐹)‘𝑥))
115fveq1d 6334 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥) = (𝐻𝑥))
1210, 11reseq12d 5535 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (((2nd𝑓)‘𝑥) ↾ (𝑥)) = (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))
136, 12mpteq12dv 4865 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥))) = (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥))))
148, 13opeq12d 4545 . 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)
16 elex 3361 . . 3 (𝐹𝑉𝐹 ∈ V)
1715, 16syl 17 . 2 (𝜑𝐹 ∈ V)
18 resfval.d . . 3 (𝜑𝐻𝑊)
19 elex 3361 . . 3 (𝐻𝑊𝐻 ∈ V)
2018, 19syl 17 . 2 (𝜑𝐻 ∈ V)
21 opex 5060 . . 3 ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))⟩ ∈ V
2221a1i 11 . 2 (𝜑 → ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))⟩ ∈ V)
232, 14, 17, 20, 22ovmpt2d 6934 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 382   = wceq 1630  wcel 2144  Vcvv 3349  cop 4320  cmpt 4861  dom cdm 5249  cres 5251  cfv 6031  (class class class)co 6792  cmpt2 6794  1st c1st 7312  2nd c2nd 7313  f cresf 16723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-resf 16727
This theorem is referenced by:  resfval2  16759  resf1st  16760  resf2nd  16761  funcres  16762
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