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Theorem resf1st 17879

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 → (1^st ‘(𝐹 ↾_f 𝐻)) = ((1^st ‘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 17877	. . 3 ⊢ (𝜑 → (𝐹 ↾_f 𝐻) = ⟨((1^st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2^nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
4	3	fveq2d 6901	. 2 ⊢ (𝜑 → (1^st ‘(𝐹 ↾_f 𝐻)) = (1^st ‘⟨((1^st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2^nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
5		fvex 6910	. . . 4 ⊢ (1^st ‘𝐹) ∈ V
6	5	resex 6033	. . 3 ⊢ ((1^st ‘𝐹) ↾ dom dom 𝐻) ∈ V
7		dmexg 7909	. . . 4 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
8		mptexg 7233	. . . 4 ⊢ (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2^nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
9	2, 7, 8	3syl 18	. . 3 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2^nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
10		op1stg 8005	. . 3 ⊢ ((((1^st ‘𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2^nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) → (1^st ‘⟨((1^st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2^nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩) = ((1^st ‘𝐹) ↾ dom dom 𝐻))
11	6, 9, 10	sylancr 586	. 2 ⊢ (𝜑 → (1^st ‘⟨((1^st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2^nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩) = ((1^st ‘𝐹) ↾ dom dom 𝐻))
12		resf1st.s	. . . . . 6 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
13	12	fndmd 6659	. . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
14	13	dmeqd 5908	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
15		dmxpid 5932	. . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆
16	14, 15	eqtrdi 2784	. . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
17	16	reseq2d 5985	. 2 ⊢ (𝜑 → ((1^st ‘𝐹) ↾ dom dom 𝐻) = ((1^st ‘𝐹) ↾ 𝑆))
18	4, 11, 17	3eqtrd 2772	1 ⊢ (𝜑 → (1^st ‘(𝐹 ↾_f 𝐻)) = ((1^st ‘𝐹) ↾ 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⟨cop 4635 ↦ cmpt 5231 × cxp 5676 dom cdm 5678 ↾ cres 5680 Fn wfn 6543 ‘cfv 6548 (class class class)co 7420 1^st c1st 7991 2^nd c2nd 7992 ↾_f cresf 17842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-resf 17846

This theorem is referenced by: (None)

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