Theorem ovmpomptd 50187

Description: The outer value of an iterated maps-to function is the corresponding inner maps-to function. (Contributed by Zhi Wang, 14-Sep-2026.)

Hypotheses

Ref	Expression
fnmpompt.l	⊢ 𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)))
fvmpomptd.u	⊢ (𝜑 → 𝑈 ∈ 𝐵)
fvmpomptd.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)

Assertion

Ref	Expression
ovmpomptd	⊢ (𝜑 → (𝑈𝐿𝑉) = (𝑎 ∈ (𝑈𝐻𝑉) ↦ (𝐴‘𝑎)))

Distinct variable groups:   𝐵,𝑎,𝑢,𝑣   𝑢,𝐴,𝑣   𝐻,𝑎,𝑢,𝑣   𝑈,𝑎,𝑢,𝑣   𝑉,𝑎,𝑢,𝑣   𝜑,𝑎,𝑢,𝑣

Allowed substitution hints:   𝐴(𝑎)   𝐿(𝑣, 𝑢, 𝑎)

Proof of Theorem ovmpomptd

Step	Hyp	Ref	Expression
1		fnmpompt.l	. . 3 ⊢ 𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)))
2	1	a1i 11	. 2 ⊢ (𝜑 → 𝐿 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎))))
3		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑈 ∧ 𝑣 = 𝑉)) → 𝑢 = 𝑈)
4		simprr 785	. . . 4 ⊢ ((𝜑 ∧ (𝑢 = 𝑈 ∧ 𝑣 = 𝑉)) → 𝑣 = 𝑉)
5	3, 4	oveq12d 7461	. . 3 ⊢ ((𝜑 ∧ (𝑢 = 𝑈 ∧ 𝑣 = 𝑉)) → (𝑢𝐻𝑣) = (𝑈𝐻𝑉))
6	5	mpteq1d 5198	. 2 ⊢ ((𝜑 ∧ (𝑢 = 𝑈 ∧ 𝑣 = 𝑉)) → (𝑎 ∈ (𝑢𝐻𝑣) ↦ (𝐴‘𝑎)) = (𝑎 ∈ (𝑈𝐻𝑉) ↦ (𝐴‘𝑎)))
7		fvmpomptd.u	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝐵)
8		fvmpomptd.v	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝐵)
9		ovex 7476	. . . 4 ⊢ (𝑈𝐻𝑉) ∈ V
10	9	mptex 7252	. . 3 ⊢ (𝑎 ∈ (𝑈𝐻𝑉) ↦ (𝐴‘𝑎)) ∈ V
11	10	a1i 11	. 2 ⊢ (𝜑 → (𝑎 ∈ (𝑈𝐻𝑉) ↦ (𝐴‘𝑎)) ∈ V)
12	2, 6, 7, 8, 11	ovmpod 7596	1 ⊢ (𝜑 → (𝑈𝐿𝑉) = (𝑎 ∈ (𝑈𝐻𝑉) ↦ (𝐴‘𝑎)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ↦ cmpt 5189  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448

This theorem is used by:  fvmpomptd  50188  parmapfunc2  50979  parmapfunc  50980