Theorem resf2nd 17911

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 7395	. 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 17908	. . . . 5 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
5	4	fveq2d 6867	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
6		fvex 6876	. . . . . 6 ⊢ (1st ‘𝐹) ∈ V
7	6	resex 6013	. . . . 5 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
8		dmexg 7878	. . . . . 6 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
9		mptexg 7201	. . . . . 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 7979	. . . . 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 596	. . . 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 2796	. . 3 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧))))
14		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → 𝑧 = ⟨𝑋, 𝑌⟩)
15	14	fveq2d 6867	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩))
16		df-ov 7395	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩)
17	15, 16	eqtr4di 2814	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = (𝑋(2nd ‘𝐹)𝑌))
18	14	fveq2d 6867	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19		df-ov 7395	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
20	18, 19	eqtr4di 2814	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌))
21	17, 20	reseq12d 5964	. . 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 5684	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
25		resf1st.s	. . . . 5 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
26	25	fndmd 6622	. . . 4 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
27	24, 26	eleqtrrd 2864	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
28		ovex 7425	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) ∈ V
29	28	resex 6013	. . . 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 6979	. 2 ⊢ (𝜑 → ((2nd ‘(𝐹 ↾f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
32	1, 31	eqtrid 2808	1 ⊢ (𝜑 → (𝑋(2nd ‘(𝐹 ↾f 𝐻))𝑌) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453  ⟨cop 4587   ↦ cmpt 5180   × cxp 5643  dom cdm 5645   ↾ cres 5647   Fn wfn 6512  ‘cfv 6517  (class class class)co 7392  1^st c1st 7964  2^nd c2nd 7965   ↾_f cresf 17873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-2nd 7967  df-resf 17877

This theorem is referenced by:  funcres  17912