Theorem resf1st 17952

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 17950	. . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
4	3	fveq2d 6887	. 2 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = (1st ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
5		fvex 6896	. . . 4 ⊢ (1st ‘𝐹) ∈ V
6	5	resex 6030	. . 3 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
7		dmexg 7899	. . . 4 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
8		mptexg 7221	. . . 4 ⊢ (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
9	2, 7, 8	3syl 19	. . 3 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
10		op1stg 7999	. . 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 598	. 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 6642	. . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
14	13	dmeqd 5897	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
15		dmxpid 5922	. . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆
16	14, 15	eqtrdi 2814	. . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
17	16	reseq2d 5980	. 2 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻) = ((1st ‘𝐹) ↾ 𝑆))
18	4, 11, 17	3eqtrd 2802	1 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4596   ↦ cmpt 5193   × cxp 5661  dom cdm 5663   ↾ cres 5665   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412  1^st c1st 7985  2^nd c2nd 7986   ↾_f cresf 17915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-resf 17919

This theorem is referenced by: (None)