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Theorem resfval 16533
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 16502 . . 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 793 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → 𝑓 = 𝐹)
43fveq2d 6182 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (1st𝑓) = (1st𝐹))
5 simprr 795 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → = 𝐻)
65dmeqd 5315 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom = dom 𝐻)
76dmeqd 5315 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → dom dom = dom dom 𝐻)
84, 7reseq12d 5386 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((1st𝑓) ↾ dom dom ) = ((1st𝐹) ↾ dom dom 𝐻))
93fveq2d 6182 . . . . . 6 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (2nd𝑓) = (2nd𝐹))
109fveq1d 6180 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → ((2nd𝑓)‘𝑥) = ((2nd𝐹)‘𝑥))
115fveq1d 6180 . . . . 5 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥) = (𝐻𝑥))
1210, 11reseq12d 5386 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (((2nd𝑓)‘𝑥) ↾ (𝑥)) = (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥)))
136, 12mpteq12dv 4724 . . 3 ((𝜑 ∧ (𝑓 = 𝐹 = 𝐻)) → (𝑥 ∈ dom ↦ (((2nd𝑓)‘𝑥) ↾ (𝑥))) = (𝑥 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑥) ↾ (𝐻𝑥))))
148, 13opeq12d 4401 . 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 3207 . . 3 (𝐹𝑉𝐹 ∈ V)
1715, 16syl 17 . 2 (𝜑𝐹 ∈ V)
18 resfval.d . . 3 (𝜑𝐻𝑊)
19 elex 3207 . . 3 (𝐻𝑊𝐻 ∈ V)
2018, 19syl 17 . 2 (𝜑𝐻 ∈ V)
21 opex 4923 . . 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 6773 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 384   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174  cmpt 4720  dom cdm 5104  cres 5106  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  f cresf 16498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-res 5116  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-resf 16502
This theorem is referenced by:  resfval2  16534  resf1st  16535  resf2nd  16536  funcres  16537
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