Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  briunov2 Structured version   Visualization version   GIF version

Theorem briunov2 40313
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. (Contributed by RP, 1-Jun-2020.)
Hypothesis
Ref Expression
briunov2.def 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
Assertion
Ref Expression
briunov2 ((𝑅𝑈𝑁𝑉) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))
Distinct variable groups:   𝑛,𝑟,𝐶,𝑁,   𝑅,𝑛,𝑟   𝑛,𝑋   𝑛,𝑌
Allowed substitution hints:   𝑈(𝑛,𝑟)   𝑉(𝑛,𝑟)   𝑋(𝑟)   𝑌(𝑟)

Proof of Theorem briunov2
StepHypRef Expression
1 briunov2.def . . 3 𝐶 = (𝑟 ∈ V ↦ 𝑛𝑁 (𝑟 𝑛))
21eliunov2 40310 . 2 ((𝑅𝑈𝑁𝑉) → (⟨𝑋, 𝑌⟩ ∈ (𝐶𝑅) ↔ ∃𝑛𝑁𝑋, 𝑌⟩ ∈ (𝑅 𝑛)))
3 df-br 5043 . 2 (𝑋(𝐶𝑅)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝐶𝑅))
4 df-br 5043 . . 3 (𝑋(𝑅 𝑛)𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ (𝑅 𝑛))
54rexbii 3235 . 2 (∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌 ↔ ∃𝑛𝑁𝑋, 𝑌⟩ ∈ (𝑅 𝑛))
62, 3, 53bitr4g 317 1 ((𝑅𝑈𝑁𝑉) → (𝑋(𝐶𝑅)𝑌 ↔ ∃𝑛𝑁 𝑋(𝑅 𝑛)𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wrex 3131  Vcvv 3469  cop 4545   ciun 4894   class class class wbr 5042  cmpt 5122  cfv 6334  (class class class)co 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143
This theorem is referenced by:  brmptiunrelexpd  40314  brtrclrec  40327  brrtrclrec  40328  briunov2uz  40329
  Copyright terms: Public domain W3C validator