Theorem resf1st 17852

Description: Value of the functor restriction operator on objects. (Contributed by Mario Carneiro, 6-Jan-2017.)

Hypotheses

Ref	Expression
resf1st.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
resf1st.h	⊢ (𝜑 → 𝐻 ∈ 𝑊)
resf1st.s	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))

Assertion

Ref	Expression
resf1st	⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆))

Proof of Theorem resf1st

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resf1st.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
2		resf1st.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
3	1, 2	resfval 17850	. . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
4	3	fveq2d 6831	. 2 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = (1st ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
5		fvex 6840	. . . 4 ⊢ (1st ‘𝐹) ∈ V
6	5	resex 5981	. . 3 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
7		dmexg 7841	. . . 4 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
8		mptexg 7165	. . . 4 ⊢ (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
9	2, 7, 8	3syl 18	. . 3 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
10		op1stg 7943	. . 3 ⊢ ((((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) → (1st ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩) = ((1st ‘𝐹) ↾ dom dom 𝐻))
11	6, 9, 10	sylancr 593	. 2 ⊢ (𝜑 → (1st ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩) = ((1st ‘𝐹) ↾ dom dom 𝐻))
12		resf1st.s	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
13	12	fndmd 6590	. . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
14	13	dmeqd 5847	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
15		dmxpid 5872	. . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆
16	14, 15	eqtrdi 2790	. . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
17	16	reseq2d 5931	. 2 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻) = ((1st ‘𝐹) ↾ 𝑆))
18	4, 11, 17	3eqtrd 2778	1 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561   ↦ cmpt 5153   × cxp 5616  dom cdm 5618   ↾ cres 5620   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930   ↾_f cresf 17815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-resf 17819

This theorem is referenced by: (None)