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Theorem eubrdm 46341
Description: If there is a unique set which is related to a class, then the class is an element of the domain of the relation. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
eubrdm (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ dom 𝑅)
Distinct variable groups:   𝐴,𝑏   𝑅,𝑏

Proof of Theorem eubrdm
StepHypRef Expression
1 eubrv 46340 . 2 (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ V)
2 iotaex 6515 . . 3 (℩𝑏𝐴𝑅𝑏) ∈ V
32a1i 11 . 2 (∃!𝑏 𝐴𝑅𝑏 → (℩𝑏𝐴𝑅𝑏) ∈ V)
4 iota4 6523 . . 3 (∃!𝑏 𝐴𝑅𝑏[(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏)
5 sbcbr12g 5198 . . . . 5 ((℩𝑏𝐴𝑅𝑏) ∈ V → ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏(℩𝑏𝐴𝑅𝑏) / 𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏) / 𝑏𝑏))
62, 5ax-mp 5 . . . 4 ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏(℩𝑏𝐴𝑅𝑏) / 𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏) / 𝑏𝑏)
7 csbconstg 3908 . . . . . 6 ((℩𝑏𝐴𝑅𝑏) ∈ V → (℩𝑏𝐴𝑅𝑏) / 𝑏𝐴 = 𝐴)
82, 7ax-mp 5 . . . . 5 (℩𝑏𝐴𝑅𝑏) / 𝑏𝐴 = 𝐴
92csbvargi 4428 . . . . 5 (℩𝑏𝐴𝑅𝑏) / 𝑏𝑏 = (℩𝑏𝐴𝑅𝑏)
108, 9breq12i 5151 . . . 4 ((℩𝑏𝐴𝑅𝑏) / 𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏) / 𝑏𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏))
116, 10sylbb 218 . . 3 ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏))
124, 11syl 17 . 2 (∃!𝑏 𝐴𝑅𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏))
13 breldmg 5906 . 2 ((𝐴 ∈ V ∧ (℩𝑏𝐴𝑅𝑏) ∈ V ∧ 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) → 𝐴 ∈ dom 𝑅)
141, 3, 12, 13syl3anc 1369 1 (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  ∃!weu 2557  Vcvv 3469  [wsbc 3774  csb 3889   class class class wbr 5142  dom cdm 5672  cio 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-dm 5682  df-iota 6494
This theorem is referenced by:  dfafv2  46435
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