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Theorem resf1st 17945
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 17943 . . 3 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
43fveq2d 6911 . 2 (𝜑 → (1st ‘(𝐹f 𝐻)) = (1st ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
5 fvex 6920 . . . 4 (1st𝐹) ∈ V
65resex 6049 . . 3 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
7 dmexg 7924 . . . 4 (𝐻𝑊 → dom 𝐻 ∈ V)
8 mptexg 7241 . . . 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 8025 . . 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 587 . 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 6674 . . . . 5 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1413dmeqd 5919 . . . 4 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
15 dmxpid 5944 . . . 4 dom (𝑆 × 𝑆) = 𝑆
1614, 15eqtrdi 2791 . . 3 (𝜑 → dom dom 𝐻 = 𝑆)
1716reseq2d 6000 . 2 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻) = ((1st𝐹) ↾ 𝑆))
184, 11, 173eqtrd 2779 1 (𝜑 → (1st ‘(𝐹f 𝐻)) = ((1st𝐹) ↾ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cmpt 5231   × cxp 5687  dom cdm 5689  cres 5691   Fn wfn 6558  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  f cresf 17908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-resf 17912
This theorem is referenced by: (None)
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