Theorem resf1st 16601

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 16599	. . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
4	3	fveq2d 6233	. 2 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = (1st ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
5		fvex 6239	. . . 4 ⊢ (1st ‘𝐹) ∈ V
6	5	resex 5478	. . 3 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
7		dmexg 7139	. . . 4 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
8		mptexg 6525	. . . 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 7222	. . 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 696	. 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		fndm 6028	. . . . . 6 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
14	12, 13	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
15	14	dmeqd 5358	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
16		dmxpid 5377	. . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆
17	15, 16	syl6eq 2701	. . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
18	17	reseq2d 5428	. 2 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻) = ((1st ‘𝐹) ↾ 𝑆))
19	4, 11, 18	3eqtrd 2689	1 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆))

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   ↦ cmpt 4762   × cxp 5141  dom cdm 5143   ↾ cres 5145   Fn wfn 5921  ‘cfv 5926  (class class class)co 6690  1^st c1st 7208  2^nd c2nd 7209   ↾_f cresf 16564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-resf 16568

This theorem is referenced by: (None)