Theorem eliunov2 39421

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 14276. (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 3428	. . . . 5 ⊢ (𝑅 ∈ 𝑈 → 𝑅 ∈ V)
2	1	adantr 473	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → 𝑅 ∈ V)
3		simpr 477	. . . . 5 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ 𝑉)
4		ovex 7007	. . . . . 6 ⊢ (𝑅 ↑ 𝑛) ∈ V
5	4	rgenw 3095	. . . . 5 ⊢ ∀𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V
6		iunexg 7475	. . . . 5 ⊢ ((𝑁 ∈ 𝑉 ∧ ∀𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V) → ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V)
7	3, 5, 6	sylancl 578	. . . 4 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V)
8		oveq1 6982	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑟 ↑ 𝑛) = (𝑅 ↑ 𝑛))
9	8	iuneq2d 4817	. . . . 5 ⊢ (𝑟 = 𝑅 → ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
10		eqid 2773	. . . . 5 ⊢ (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
11	9, 10	fvmptg 6592	. . . 4 ⊢ ((𝑅 ∈ V ∧ ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛) ∈ V) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
12	2, 7, 11	syl2anc 576	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅) = ∪ 𝑛 ∈ 𝑁 (𝑅 ↑ 𝑛))
13		eleq2 2849	. . . 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 4793	. . . . 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 502	. . 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 676	. 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 6496	. . . . . 6 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (𝐶‘𝑅) = ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅))
20	19	eleq2d 2846	. . . . 5 ⊢ (𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛)) → (𝑋 ∈ (𝐶‘𝑅) ↔ 𝑋 ∈ ((𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))‘𝑅)))
21	20	bibi1d 336	. . . 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 333	. . 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 223	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ 𝑉) → (𝑋 ∈ (𝐶‘𝑅) ↔ ∃𝑛 ∈ 𝑁 𝑋 ∈ (𝑅 ↑ 𝑛)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051  ∀wral 3083  ∃wrex 3084  Vcvv 3410  ∪ ciun 4789   ↦ cmpt 5005  ‘cfv 6186  (class class class)co 6975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-rep 5046  ax-sep 5057  ax-nul 5064  ax-pr 5183  ax-un 7278

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-reu 3090  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-ov 6978

This theorem is referenced by:  eltrclrec  39422  elrtrclrec  39423  briunov2  39424  eliunov2uz  39441  ov2ssiunov2  39442