Theorem resf1st 17830

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 17828	. . 3 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
4	3	fveq2d 6846	. 2 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = (1st ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
5		fvex 6855	. . . 4 ⊢ (1st ‘𝐹) ∈ V
6	5	resex 5996	. . 3 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
7		dmexg 7853	. . . 4 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
8		mptexg 7177	. . . 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 7955	. . 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 588	. 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 6605	. . . . 5 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
14	13	dmeqd 5862	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
15		dmxpid 5887	. . . 4 ⊢ dom (𝑆 × 𝑆) = 𝑆
16	14, 15	eqtrdi 2788	. . 3 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
17	16	reseq2d 5946	. 2 ⊢ (𝜑 → ((1st ‘𝐹) ↾ dom dom 𝐻) = ((1st ‘𝐹) ↾ 𝑆))
18	4, 11, 17	3eqtrd 2776	1 ⊢ (𝜑 → (1st ‘(𝐹 ↾f 𝐻)) = ((1st ‘𝐹) ↾ 𝑆))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588   ↦ cmpt 5181   × cxp 5630  dom cdm 5632   ↾ cres 5634   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942   ↾_f cresf 17793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-resf 17797

This theorem is referenced by: (None)