Theorem eliunov2 43836

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 14972. (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 2733	. . . 4 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
2		oveq1 7362	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ↑ 𝑛) = (𝑅 ↑ 𝑛))
3	2	iuneq2d 4974	. . . 4 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
4		elex 3458	. . . . 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 7388	. . . . . 6 ⊢ (𝑅 ↑ 𝑛) ∈ V
8	7	rgenw 3052	. . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V
9		iunexg 7904	. . . . 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 6961	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
12		eleq2 2822	. . . 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 4947	. . . . 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 6830	. . . . . 6 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (𝐶‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅))
19	18	eleq2d 2819	. . . . 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 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  Vcvv 3437  ∪ ciun 4943   ↦ cmpt 5176  ‘cfv 6489  (class class class)co 7355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358

This theorem is referenced by:  eltrclrec  43837  elrtrclrec  43838  briunov2  43839  eliunov2uz  43856  ov2ssiunov2  43857