Theorem resf2nd 17659

Description: Value of the functor restriction operator on morphisms. (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))
resf2nd.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
resf2nd.y	⊢ (𝜑 → 𝑌 ∈ 𝑆)

Assertion

Ref	Expression
resf2nd	⊢ (𝜑 → (𝑋(2nd ‘(𝐹 ↾f 𝐻))𝑌) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))

Proof of Theorem resf2nd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7310	. 2 ⊢ (𝑋(2nd ‘(𝐹 ↾f 𝐻))𝑌) = ((2nd ‘(𝐹 ↾f 𝐻))‘⟨𝑋, 𝑌⟩)
2		resf1st.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
3		resf1st.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ 𝑊)
4	2, 3	resfval 17656	. . . . 5 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
5	4	fveq2d 6808	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
6		fvex 6817	. . . . . 6 ⊢ (1st ‘𝐹) ∈ V
7	6	resex 5951	. . . . 5 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
8		dmexg 7782	. . . . . 6 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
9		mptexg 7129	. . . . . 6 ⊢ (dom 𝐻 ∈ V → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
10	3, 8, 9	3syl 18	. . . . 5 ⊢ (𝜑 → (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V)
11		op2ndg 7876	. . . . 5 ⊢ ((((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V ∧ (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))) ∈ V) → (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
12	7, 10, 11	sylancr 588	. . . 4 ⊢ (𝜑 → (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
13	5, 12	eqtrd 2776	. . 3 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
14		simpr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
15	14	fveq2d 6808	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩))
16		df-ov 7310	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩)
17	15, 16	eqtr4di 2794	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = (𝑋(2nd ‘𝐹)𝑌))
18	14	fveq2d 6808	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19		df-ov 7310	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
20	18, 19	eqtr4di 2794	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌))
21	17, 20	reseq12d 5904	. . 3 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
22		resf2nd.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
23		resf2nd.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
24	22, 23	opelxpd 5638	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
25		resf1st.s	. . . . 5 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
26	25	fndmd 6569	. . . 4 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
27	24, 26	eleqtrrd 2840	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
28		ovex 7340	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) ∈ V
29	28	resex 5951	. . . 4 ⊢ ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V
30	29	a1i 11	. . 3 ⊢ (𝜑 → ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)) ∈ V)
31	13, 21, 27, 30	fvmptd 6914	. 2 ⊢ (𝜑 → ((2nd ‘(𝐹 ↾f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
32	1, 31	eqtrid 2788	1 ⊢ (𝜑 → (𝑋(2nd ‘(𝐹 ↾f 𝐻))𝑌) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  Vcvv 3437  ⟨cop 4571   ↦ cmpt 5164   × cxp 5598  dom cdm 5600   ↾ cres 5602   Fn wfn 6453  ‘cfv 6458  (class class class)co 7307  1^st c1st 7861  2^nd c2nd 7862   ↾_f cresf 17621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-2nd 7864  df-resf 17625

This theorem is referenced by:  funcres  17660