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Theorem resf1st 17785
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 17783 . . 3 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
43fveq2d 6847 . 2 (𝜑 → (1st ‘(𝐹f 𝐻)) = (1st ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
5 fvex 6856 . . . 4 (1st𝐹) ∈ V
65resex 5986 . . 3 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
7 dmexg 7841 . . . 4 (𝐻𝑊 → dom 𝐻 ∈ V)
8 mptexg 7172 . . . 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 7934 . . 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 588 . 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))
1312fndmd 6608 . . . . 5 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1413dmeqd 5862 . . . 4 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
15 dmxpid 5886 . . . 4 dom (𝑆 × 𝑆) = 𝑆
1614, 15eqtrdi 2789 . . 3 (𝜑 → dom dom 𝐻 = 𝑆)
1716reseq2d 5938 . 2 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻) = ((1st𝐹) ↾ 𝑆))
184, 11, 173eqtrd 2777 1 (𝜑 → (1st ‘(𝐹f 𝐻)) = ((1st𝐹) ↾ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3444  cop 4593  cmpt 5189   × cxp 5632  dom cdm 5634  cres 5636   Fn wfn 6492  cfv 6497  (class class class)co 7358  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-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  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-iun 4957  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-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-resf 17752
This theorem is referenced by: (None)
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