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Theorem resf1st 16775
 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 16773 . . 3 (𝜑 → (𝐹f 𝐻) = ⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩)
43fveq2d 6357 . 2 (𝜑 → (1st ‘(𝐹f 𝐻)) = (1st ‘⟨((1st𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd𝐹)‘𝑧) ↾ (𝐻𝑧)))⟩))
5 fvex 6363 . . . 4 (1st𝐹) ∈ V
65resex 5601 . . 3 ((1st𝐹) ↾ dom dom 𝐻) ∈ V
7 dmexg 7263 . . . 4 (𝐻𝑊 → dom 𝐻 ∈ V)
8 mptexg 6649 . . . 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 7346 . . 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 698 . 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 6151 . . . . . 6 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
1412, 13syl 17 . . . . 5 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1514dmeqd 5481 . . . 4 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
16 dmxpid 5500 . . . 4 dom (𝑆 × 𝑆) = 𝑆
1715, 16syl6eq 2810 . . 3 (𝜑 → dom dom 𝐻 = 𝑆)
1817reseq2d 5551 . 2 (𝜑 → ((1st𝐹) ↾ dom dom 𝐻) = ((1st𝐹) ↾ 𝑆))
194, 11, 183eqtrd 2798 1 (𝜑 → (1st ‘(𝐹f 𝐻)) = ((1st𝐹) ↾ 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327   ↦ cmpt 4881   × cxp 5264  dom cdm 5266   ↾ cres 5268   Fn wfn 6044  ‘cfv 6049  (class class class)co 6814  1st c1st 7332  2nd c2nd 7333   ↾f cresf 16738 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-resf 16742 This theorem is referenced by: (None)
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