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

Theorem resfval 16318
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 16287 . . 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 789 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → 𝑓 = 𝐹)
43fveq2d 6089 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (1st𝑓) = (1st𝐹))
5 simprr 791 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → = 𝐻)
65dmeqd 5232 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom = dom 𝐻)
76dmeqd 5232 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom dom = dom dom 𝐻)
84, 7reseq12d 5302 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((1st𝑓) ↾ dom dom ) = ((1st𝐹) ↾ dom dom 𝐻))
93fveq2d 6089 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (2nd𝑓) = (2nd𝐹))
109fveq1d 6087 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((2nd𝑓)‘𝑥) = ((2nd𝐹)‘𝑥))
115fveq1d 6087 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥) = (𝐻𝑥))
1210, 11reseq12d 5302 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (((2nd𝑓)‘𝑥) ↾ (𝑥)) = (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))
136, 12mpteq12dv 4654 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥))) = (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥))))
148, 13opeq12d 4339 . 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 3181 . . 3 (𝐹𝑉𝐹 ∈ V)
1715, 16syl 17 . 2 (𝜑𝐹 ∈ V)
18 resfval.d . . 3 (𝜑𝐻𝑊)
19 elex 3181 . . 3 (𝐻𝑊𝐻 ∈ V)
2018, 19syl 17 . 2 (𝜑𝐻 ∈ V)
21 opex 4850 . . 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 6661 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 1474  wcel 1976  Vcvv 3169  cop 4127  cmpt 4634  dom cdm 5025  cres 5027  cfv 5787  (class class class)co 6524  cmpt2 6526  1st c1st 7031  2nd c2nd 7032  f cresf 16283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-res 5037  df-iota 5751  df-fun 5789  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-resf 16287
This theorem is referenced by:  resfval2  16319  resf1st  16320  resf2nd  16321  funcres  16322
  Copyright terms: Public domain W3C validator