Theorem eliunov2 36990

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 the element is a member of that operator value. Generalized from dfrtrclrec2 13645. (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		elex 3185	. . . . 5 ⊢ (𝑅 ∈ 𝑈 → 𝑅 ∈ V)
2	1	adantr 480	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → 𝑅 ∈ V)
3		simpr 476	. . . . 5 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
4		ovex 6577	. . . . . 6 ⊢ (𝑅 ↑ 𝑛) ∈ V
5	4	rgenw 2908	. . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V
6		iunexg 7035	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V) → ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V)
7	3, 5, 6	sylancl 693	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V)
8		oveq1 6556	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 ↑ 𝑛) = (𝑅 ↑ 𝑛))
9	8	iuneq2d 4483	. . . . 5 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
10		eqid 2610	. . . . 5 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
11	9, 10	fvmptg 6189	. . . 4 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
12	2, 7, 11	syl2anc 691	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
13		eleq2 2677	. . . 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)))
14		eliun 4460	. . . . 5 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛))
15	14	a1i 11	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))
16	13, 15	sylan9bb 732	. . 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)))
17	12, 16	mpancom 700	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))
18		mptiunov2.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
19		fveq1 6102	. . . . . 6 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (𝐶‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅))
20	19	eleq2d 2673	. . . . 5 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (𝑋 ∈ (𝐶‘𝑅) ↔ 𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅)))
21	20	bibi1d 332	. . . 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))))
22	21	imbi2d 329	. . 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)))))
23	18, 22	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))))
24	17, 23	mpbir 220	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ∪ ciun 4455   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552

This theorem is referenced by:  eltrclrec  36991  elrtrclrec  36992  briunov2  36993  eliunov2uz  37010  ov2ssiunov2  37011