Theorem eliunov2 41287

Description: Membership in the indexed union over operator values where the index varies the second input is equivalent to the existence of at least one index such that the element is a member of that operator value. Generalized from dfrtrclrec2 14769. (Contributed by RP, 1-Jun-2020.)

Hypothesis

Ref	Expression
mptiunov2.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))

Assertion

Ref	Expression
eliunov2	⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))

Distinct variable groups:   𝑛,𝑟,𝐶,𝑁, ↑   𝑅,𝑛,𝑟   𝑛,𝑋

Allowed substitution hints:   𝑈(𝑛,𝑟)   𝑉(𝑛,𝑟)   𝑋(𝑟)

Proof of Theorem eliunov2

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
2		oveq1 7282	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ↑ 𝑛) = (𝑅 ↑ 𝑛))
3	2	iuneq2d 4953	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
4		elex 3450	. . . . 5 ⊢ (𝑅 ∈ 𝑈 → 𝑅 ∈ V)
5	4	adantr 481	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → 𝑅 ∈ V)
6		simpr 485	. . . . 5 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
7		ovex 7308	. . . . . 6 ⊢ (𝑅 ↑ 𝑛) ∈ V
8	7	rgenw 3076	. . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V
9		iunexg 7806	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V) → ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V)
10	6, 8, 9	sylancl 586	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V)
11	1, 3, 5, 10	fvmptd3 6898	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
12		eleq2 2827	. . . 4 ⊢ (((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) → (𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) ↔ 𝑋 ∈ ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛)))
13		eliun 4928	. . . . 5 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛))
14	13	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))
15	12, 14	sylan9bb 510	. . 3 ⊢ ((((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∧ (𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉)) → (𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))
16	11, 15	mpancom 685	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))
17		mptiunov2.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
18		fveq1 6773	. . . . . 6 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (𝐶‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅))
19	18	eleq2d 2824	. . . . 5 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (𝑋 ∈ (𝐶‘𝑅) ↔ 𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅)))
20	19	bibi1d 344	. . . 4 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → ((𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)) ↔ (𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛))))
21	20	imbi2d 341	. . 3 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛))) ↔ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))))
22	17, 21	ax-mp 5	. 2 ⊢ (((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛))) ↔ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛))))
23	16, 22	mpbir 230	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432  ∪ ciun 4924   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278

This theorem is referenced by:  eltrclrec  41288  elrtrclrec  41289  briunov2  41290  eliunov2uz  41307  ov2ssiunov2  41308