Theorem ixpovf 50288

Description: A member of an indexed product of function spaces has the indicated function type at each pair of indices. Hom-functions of a functor are one application. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
ixpovf.p	⊢ (𝜑 → 𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))𝑁(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))
ixpovf.u	⊢ (𝜑 → 𝑈 ∈ 𝑊)
ixpovf.v	⊢ (𝜑 → 𝑉 ∈ 𝑊)

Assertion

Ref	Expression
ixpovf	⊢ (𝜑 → (𝑈𝑃𝑉):(𝑈𝐻𝑉)⟶((𝑂‘𝑈)𝑁(𝑂‘𝑉)))

Distinct variable groups:   𝑧,𝐻   𝑧,𝑁   𝑧,𝑂   𝑧,𝑃   𝑧,𝑈   𝑧,𝑉   𝑧,𝑊

Allowed substitution hint:   𝜑(𝑧)

Proof of Theorem ixpovf

Step	Hyp	Ref	Expression
1		ixpovf.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))𝑁(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))
2		ixpovf.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑊)
3		ixpovf.v	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝑊)
4	2, 3	opelxpd 5689	. . . 4 ⊢ (𝜑 → ⟨𝑈, 𝑉⟩ ∈ (𝑊 × 𝑊))
5		2fveq3 6890	. . . . . . 7 ⊢ (𝑧 = ⟨𝑈, 𝑉⟩ → (𝑂‘(1st ‘𝑧)) = (𝑂‘(1st ‘⟨𝑈, 𝑉⟩)))
6		2fveq3 6890	. . . . . . 7 ⊢ (𝑧 = ⟨𝑈, 𝑉⟩ → (𝑂‘(2nd ‘𝑧)) = (𝑂‘(2nd ‘⟨𝑈, 𝑉⟩)))
7	5, 6	oveq12d 7438	. . . . . 6 ⊢ (𝑧 = ⟨𝑈, 𝑉⟩ → ((𝑂‘(1st ‘𝑧))𝑁(𝑂‘(2nd ‘𝑧))) = ((𝑂‘(1st ‘⟨𝑈, 𝑉⟩))𝑁(𝑂‘(2nd ‘⟨𝑈, 𝑉⟩))))
8		fveq2 6885	. . . . . 6 ⊢ (𝑧 = ⟨𝑈, 𝑉⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑈, 𝑉⟩))
9	7, 8	oveq12d 7438	. . . . 5 ⊢ (𝑧 = ⟨𝑈, 𝑉⟩ → (((𝑂‘(1st ‘𝑧))𝑁(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) = (((𝑂‘(1st ‘⟨𝑈, 𝑉⟩))𝑁(𝑂‘(2nd ‘⟨𝑈, 𝑉⟩))) ↑m (𝐻‘⟨𝑈, 𝑉⟩)))
10	9	fvixp 8932	. . . 4 ⊢ ((𝑃 ∈ X𝑧 ∈ (𝑊 × 𝑊)(((𝑂‘(1st ‘𝑧))𝑁(𝑂‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ⟨𝑈, 𝑉⟩ ∈ (𝑊 × 𝑊)) → (𝑃‘⟨𝑈, 𝑉⟩) ∈ (((𝑂‘(1st ‘⟨𝑈, 𝑉⟩))𝑁(𝑂‘(2nd ‘⟨𝑈, 𝑉⟩))) ↑m (𝐻‘⟨𝑈, 𝑉⟩)))
11	1, 4, 10	syl2anc 596	. . 3 ⊢ (𝜑 → (𝑃‘⟨𝑈, 𝑉⟩) ∈ (((𝑂‘(1st ‘⟨𝑈, 𝑉⟩))𝑁(𝑂‘(2nd ‘⟨𝑈, 𝑉⟩))) ↑m (𝐻‘⟨𝑈, 𝑉⟩)))
12	11	elmaprd 8872	. 2 ⊢ (𝜑 → (𝑃‘⟨𝑈, 𝑉⟩):(𝐻‘⟨𝑈, 𝑉⟩)⟶((𝑂‘(1st ‘⟨𝑈, 𝑉⟩))𝑁(𝑂‘(2nd ‘⟨𝑈, 𝑉⟩))))
13		df-ov 7423	. . . . 5 ⊢ (𝑈𝑃𝑉) = (𝑃‘⟨𝑈, 𝑉⟩)
14	13	eqcomi 2769	. . . 4 ⊢ (𝑃‘⟨𝑈, 𝑉⟩) = (𝑈𝑃𝑉)
15	14	a1i 11	. . 3 ⊢ (𝜑 → (𝑃‘⟨𝑈, 𝑉⟩) = (𝑈𝑃𝑉))
16		df-ov 7423	. . . . 5 ⊢ (𝑈𝐻𝑉) = (𝐻‘⟨𝑈, 𝑉⟩)
17	16	eqcomi 2769	. . . 4 ⊢ (𝐻‘⟨𝑈, 𝑉⟩) = (𝑈𝐻𝑉)
18	17	a1i 11	. . 3 ⊢ (𝜑 → (𝐻‘⟨𝑈, 𝑉⟩) = (𝑈𝐻𝑉))
19	2	elexd 3473	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ V)
20	3	elexd 3473	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ V)
21		op1stg 8013	. . . . . 6 ⊢ ((𝑈 ∈ V ∧ 𝑉 ∈ V) → (1st ‘⟨𝑈, 𝑉⟩) = 𝑈)
22	19, 20, 21	syl2anc 596	. . . . 5 ⊢ (𝜑 → (1st ‘⟨𝑈, 𝑉⟩) = 𝑈)
23	22	fveq2d 6889	. . . 4 ⊢ (𝜑 → (𝑂‘(1st ‘⟨𝑈, 𝑉⟩)) = (𝑂‘𝑈))
24		op2ndg 8014	. . . . . 6 ⊢ ((𝑈 ∈ V ∧ 𝑉 ∈ V) → (2nd ‘⟨𝑈, 𝑉⟩) = 𝑉)
25	19, 20, 24	syl2anc 596	. . . . 5 ⊢ (𝜑 → (2nd ‘⟨𝑈, 𝑉⟩) = 𝑉)
26	25	fveq2d 6889	. . . 4 ⊢ (𝜑 → (𝑂‘(2nd ‘⟨𝑈, 𝑉⟩)) = (𝑂‘𝑉))
27	23, 26	oveq12d 7438	. . 3 ⊢ (𝜑 → ((𝑂‘(1st ‘⟨𝑈, 𝑉⟩))𝑁(𝑂‘(2nd ‘⟨𝑈, 𝑉⟩))) = ((𝑂‘𝑈)𝑁(𝑂‘𝑉)))
28	15, 18, 27	feq123d 6698	. 2 ⊢ (𝜑 → ((𝑃‘⟨𝑈, 𝑉⟩):(𝐻‘⟨𝑈, 𝑉⟩)⟶((𝑂‘(1st ‘⟨𝑈, 𝑉⟩))𝑁(𝑂‘(2nd ‘⟨𝑈, 𝑉⟩))) ↔ (𝑈𝑃𝑉):(𝑈𝐻𝑉)⟶((𝑂‘𝑈)𝑁(𝑂‘𝑉))))
29	12, 28	mpbid 235	1 ⊢ (𝜑 → (𝑈𝑃𝑉):(𝑈𝐻𝑉)⟶((𝑂‘𝑈)𝑁(𝑂‘𝑉)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4589   × cxp 5648  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000   ↑_m cmap 8849  Xcixp 8927

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8851  df-ixp 8928

This theorem is used by:  fucidcobi  50438