Theorem eliunov2 43668

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 15024. (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 2729	. . . 4 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
2		oveq1 7394	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ↑ 𝑛) = (𝑅 ↑ 𝑛))
3	2	iuneq2d 4986	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
4		elex 3468	. . . . 5 ⊢ (𝑅 ∈ 𝑈 → 𝑅 ∈ V)
5	4	adantr 480	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → 𝑅 ∈ V)
6		simpr 484	. . . . 5 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
7		ovex 7420	. . . . . 6 ⊢ (𝑅 ↑ 𝑛) ∈ V
8	7	rgenw 3048	. . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V
9		iunexg 7942	. . . . 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 6991	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
12		eleq2 2817	. . . 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 4959	. . . . 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 509	. . 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 688	. 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 6857	. . . . . 6 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (𝐶‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅))
19	18	eleq2d 2814	. . . . 5 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (𝑋 ∈ (𝐶‘𝑅) ↔ 𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅)))
20	19	bibi1d 343	. . . 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 340	. . 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 231	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447  ∪ ciun 4955   ↦ cmpt 5188  ‘cfv 6511  (class class class)co 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390

This theorem is referenced by:  eltrclrec  43669  elrtrclrec  43670  briunov2  43671  eliunov2uz  43688  ov2ssiunov2  43689