Theorem resf2nd 17799

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 7349	. 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 17796	. . . . 5 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
5	4	fveq2d 6826	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
6		fvex 6835	. . . . . 6 ⊢ (1st ‘𝐹) ∈ V
7	6	resex 5978	. . . . 5 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
8		dmexg 7831	. . . . . 6 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
9		mptexg 7155	. . . . . 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 7934	. . . . 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 587	. . . 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 2766	. . 3 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
14		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
15	14	fveq2d 6826	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩))
16		df-ov 7349	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩)
17	15, 16	eqtr4di 2784	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = (𝑋(2nd ‘𝐹)𝑌))
18	14	fveq2d 6826	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19		df-ov 7349	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
20	18, 19	eqtr4di 2784	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌))
21	17, 20	reseq12d 5929	. . 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 5655	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
25		resf1st.s	. . . . 5 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
26	25	fndmd 6586	. . . 4 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
27	24, 26	eleqtrrd 2834	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
28		ovex 7379	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) ∈ V
29	28	resex 5978	. . . 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 6936	. 2 ⊢ (𝜑 → ((2nd ‘(𝐹 ↾f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
32	1, 31	eqtrid 2778	1 ⊢ (𝜑 → (𝑋(2nd ‘(𝐹 ↾f 𝐻))𝑌) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4582   ↦ cmpt 5172   × cxp 5614  dom cdm 5616   ↾ cres 5618   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920   ↾_f cresf 17761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-2nd 7922  df-resf 17765

This theorem is referenced by:  funcres  17800