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Theorem resf1st 17223
 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 17221 . . 3 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
43fveq2d 6662 . 2 (𝜑 → (1st ‘(𝐹f 𝐻)) = (1st ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
5 fvex 6671 . . . 4 (1st𝐹) ∈ V
65resex 5871 . . 3 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
7 dmexg 7613 . . . 4 (𝐻𝑊 → dom 𝐻 ∈ V)
8 mptexg 6975 . . . 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 7705 . . 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 590 . 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 6438 . . . . 5 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1413dmeqd 5745 . . . 4 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
15 dmxpid 5771 . . . 4 dom (𝑆 × 𝑆) = 𝑆
1614, 15eqtrdi 2809 . . 3 (𝜑 → dom dom 𝐻 = 𝑆)
1716reseq2d 5823 . 2 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻) = ((1st𝐹) ↾ 𝑆))
184, 11, 173eqtrd 2797 1 (𝜑 → (1st ‘(𝐹f 𝐻)) = ((1st𝐹) ↾ 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4528   ↦ cmpt 5112   × cxp 5522  dom cdm 5524   ↾ cres 5526   Fn wfn 6330  ‘cfv 6335  (class class class)co 7150  1st c1st 7691  2nd c2nd 7692   ↾f cresf 17186 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-resf 17190 This theorem is referenced by: (None)
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