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Theorem resf1st 16601
Description: Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resf1st.f (𝜑𝐹𝑉)
resf1st.h (𝜑𝐻𝑊)
resf1st.s (𝜑𝐻 Fn (𝑆 × 𝑆))
Assertion
Ref Expression
resf1st (𝜑 → (1st ‘(𝐹f 𝐻)) = ((1st𝐹) ↾ 𝑆))

Proof of Theorem resf1st
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 resf1st.f . . . 4 (𝜑𝐹𝑉)
2 resf1st.h . . . 4 (𝜑𝐻𝑊)
31, 2resfval 16599 . . 3 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
43fveq2d 6233 . 2 (𝜑 → (1st ‘(𝐹f 𝐻)) = (1st ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
5 fvex 6239 . . . 4 (1st𝐹) ∈ V
65resex 5478 . . 3 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
7 dmexg 7139 . . . 4 (𝐻𝑊 → dom 𝐻 ∈ V)
8 mptexg 6525 . . . 4 (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
92, 7, 83syl 18 . . 3 (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V)
10 op1stg 7222 . . 3 ((((1st𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧))) ∈ V) → (1st ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = ((1st𝐹) ↾ dom dom 𝐻))
116, 9, 10sylancr 696 . 2 (𝜑 → (1st ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩) = ((1st𝐹) ↾ dom dom 𝐻))
12 resf1st.s . . . . . 6 (𝜑𝐻 Fn (𝑆 × 𝑆))
13 fndm 6028 . . . . . 6 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
1412, 13syl 17 . . . . 5 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1514dmeqd 5358 . . . 4 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
16 dmxpid 5377 . . . 4 dom (𝑆 × 𝑆) = 𝑆
1715, 16syl6eq 2701 . . 3 (𝜑 → dom dom 𝐻 = 𝑆)
1817reseq2d 5428 . 2 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻) = ((1st𝐹) ↾ 𝑆))
194, 11, 183eqtrd 2689 1 (𝜑 → (1st ‘(𝐹f 𝐻)) = ((1st𝐹) ↾ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cop 4216  cmpt 4762   × cxp 5141  dom cdm 5143  cres 5145   Fn wfn 5921  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  f cresf 16564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-resf 16568
This theorem is referenced by: (None)
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