Theorem resf2nd 17270

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 7173	. 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 17267	. . . . 5 ⊢ (𝜑 → (𝐹 ↾f 𝐻) = ⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩)
5	4	fveq2d 6678	. . . 4 ⊢ (𝜑 → (2nd ‘(𝐹 ↾f 𝐻)) = (2nd ‘⟨((1st ‘𝐹) ↾ dom dom 𝐻), (𝑧 ∈ dom 𝐻 ↦ (((2nd ‘𝐹)‘𝑧) ↾ (𝐻‘𝑧)))⟩))
6		fvex 6687	. . . . . 6 ⊢ (1st ‘𝐹) ∈ V
7	6	resex 5873	. . . . 5 ⊢ ((1st ‘𝐹) ↾ dom dom 𝐻) ∈ V
8		dmexg 7634	. . . . . 6 ⊢ (𝐻 ∈ 𝑊 → dom 𝐻 ∈ V)
9		mptexg 6994	. . . . . 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 7727	. . . . 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 590	. . . 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 2773	. . 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 6678	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩))
16		df-ov 7173	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) = ((2nd ‘𝐹)‘⟨𝑋, 𝑌⟩)
17	15, 16	eqtr4di 2791	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → ((2nd ‘𝐹)‘𝑧) = (𝑋(2nd ‘𝐹)𝑌))
18	14	fveq2d 6678	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
19		df-ov 7173	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
20	18, 19	eqtr4di 2791	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = ⟨𝑋, 𝑌⟩) → (𝐻‘𝑧) = (𝑋𝐻𝑌))
21	17, 20	reseq12d 5826	. . 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 5563	. . . 4 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝑆 × 𝑆))
25		resf1st.s	. . . . 5 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
26	25	fndmd 6442	. . . 4 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
27	24, 26	eleqtrrd 2836	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom 𝐻)
28		ovex 7203	. . . . 5 ⊢ (𝑋(2nd ‘𝐹)𝑌) ∈ V
29	28	resex 5873	. . . 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 6782	. 2 ⊢ (𝜑 → ((2nd ‘(𝐹 ↾f 𝐻))‘⟨𝑋, 𝑌⟩) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))
32	1, 31	syl5eq 2785	1 ⊢ (𝜑 → (𝑋(2nd ‘(𝐹 ↾f 𝐻))𝑌) = ((𝑋(2nd ‘𝐹)𝑌) ↾ (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1542   ∈ wcel 2114  Vcvv 3398  ⟨cop 4522   ↦ cmpt 5110   × cxp 5523  dom cdm 5525   ↾ cres 5527   Fn wfn 6334  ‘cfv 6339  (class class class)co 7170  1^st c1st 7712  2^nd c2nd 7713   ↾_f cresf 17232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-2nd 7715  df-resf 17236

This theorem is referenced by:  funcres  17271