MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resfval Structured version   Visualization version   GIF version

Theorem resfval 17783
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 17752 . . 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 770 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → 𝑓 = 𝐹)
43fveq2d 6847 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (1st𝑓) = (1st𝐹))
5 simprr 772 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → = 𝐻)
65dmeqd 5862 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom = dom 𝐻)
76dmeqd 5862 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom dom = dom dom 𝐻)
84, 7reseq12d 5939 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((1st𝑓) ↾ dom dom ) = ((1st𝐹) ↾ dom dom 𝐻))
93fveq2d 6847 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (2nd𝑓) = (2nd𝐹))
109fveq1d 6845 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((2nd𝑓)‘𝑥) = ((2nd𝐹)‘𝑥))
115fveq1d 6845 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥) = (𝐻𝑥))
1210, 11reseq12d 5939 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (((2nd𝑓)‘𝑥) ↾ (𝑥)) = (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))
136, 12mpteq12dv 5197 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥))) = (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥))))
148, 13opeq12d 4839 . 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 3464 . 2 (𝜑𝐹 ∈ V)
17 resfval.d . . 3 (𝜑𝐻𝑊)
1817elexd 3464 . 2 (𝜑𝐻 ∈ V)
19 opex 5422 . . 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 7508 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 397   = wceq 1542  wcel 2107  Vcvv 3444  cop 4593  cmpt 5189  dom cdm 5634  cres 5636  cfv 6497  (class class class)co 7358  cmpo 7360  1st c1st 7920  2nd c2nd 7921  f cresf 17748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-resf 17752
This theorem is referenced by:  resfval2  17784  resf1st  17785  resf2nd  17786  funcres  17787
  Copyright terms: Public domain W3C validator