Theorem briunov2uz 44142

Description: Two classes related by 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 two classes are related by that operator value. The index set 𝑁 is restricted to an upper range of integers. (Contributed by RP, 2-Jun-2020.)

Hypothesis

Ref	Expression
briunov2uz.def	⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))

Assertion

Ref	Expression
briunov2uz	⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 = (ℤ≥‘𝑀)) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))

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

Allowed substitution hints:   𝑈(𝑛,𝑟)   𝑀(𝑛,𝑟)   𝑋(𝑟)   𝑌(𝑟)

Proof of Theorem briunov2uz

Step	Hyp	Ref	Expression
1		simpr 485	. . 3 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 = (ℤ≥‘𝑀)) → 𝑁 = (ℤ≥‘𝑀))
2		fvex 6840	. . 3 ⊢ (ℤ≥‘𝑀) ∈ V
3	1, 2	eqeltrdi 2847	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 = (ℤ≥‘𝑀)) → 𝑁 ∈ V)
4		briunov2uz.def	. . 3 ⊢ 𝐶 = (𝑟 ∈ V ↦ ∪ 𝑛 ∈ 𝑁 (𝑟 ↑ 𝑛))
5	4	briunov2 44126	. 2 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 ∈ V) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))
6	3, 5	syldan 597	1 ⊢ ((𝑅 ∈ 𝑈 ∧ 𝑁 = (ℤ≥‘𝑀)) → (𝑋(𝐶‘𝑅)𝑌 ↔ ∃𝑛 ∈ 𝑁 𝑋(𝑅 ↑ 𝑛)𝑌))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  Vcvv 3431  ∪ ciun 4921   class class class wbr 5072   ↦ cmpt 5153  ‘cfv 6485  (class class class)co 7356  ℤ_≥cuz 12779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-ov 7359

This theorem is referenced by:  iunrelexpuztr  44163