Theorem resf2nd 17862

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 7370	. 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 17859	. . . . 5 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
5	4	fveq2d 6844	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
6		fvex 6853	. . . . . 6 ⊢ (1st ‘𝐹) ∈ V
7	6	resex 5994	. . . . 5 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
8		dmexg 7852	. . . . . 6 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
9		mptexg 7176	. . . . . 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 7955	. . . . 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 2771	. . 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 6844	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩))
16		df-ov 7370	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩)
17	15, 16	eqtr4di 2789	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = (𝑋(2nd ‘𝐹)𝑌))
18	14	fveq2d 6844	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19		df-ov 7370	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
20	18, 19	eqtr4di 2789	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌))
21	17, 20	reseq12d 5945	. . 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 5670	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
25		resf1st.s	. . . . 5 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
26	25	fndmd 6603	. . . 4 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
27	24, 26	eleqtrrd 2839	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
28		ovex 7400	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) ∈ V
29	28	resex 5994	. . . 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 6955	. 2 ⊢ (𝜑 → ((2nd ‘(𝐹 ↾f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
32	1, 31	eqtrid 2783	1 ⊢ (𝜑 → (𝑋(2nd ‘(𝐹 ↾f 𝐻))𝑌) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ⟨cop 4573   ↦ cmpt 5166   × cxp 5629  dom cdm 5631   ↾ cres 5633   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941   ↾_f cresf 17824

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-resf 17828

This theorem is referenced by:  funcres  17863