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Theorem eubrdm 43270
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 43269 . 2 (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ V)
2 iotaex 6334 . . 3 (℩𝑏𝐴𝑅𝑏) ∈ V
32a1i 11 . 2 (∃!𝑏 𝐴𝑅𝑏 → (℩𝑏𝐴𝑅𝑏) ∈ V)
4 iota4 6335 . . 3 (∃!𝑏 𝐴𝑅𝑏[(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏)
5 sbcbr12g 5121 . . . . 5 ((℩𝑏𝐴𝑅𝑏) ∈ V → ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏(℩𝑏𝐴𝑅𝑏) / 𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏) / 𝑏𝑏))
62, 5ax-mp 5 . . . 4 ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏(℩𝑏𝐴𝑅𝑏) / 𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏) / 𝑏𝑏)
7 csbconstg 3901 . . . . . 6 ((℩𝑏𝐴𝑅𝑏) ∈ V → (℩𝑏𝐴𝑅𝑏) / 𝑏𝐴 = 𝐴)
82, 7ax-mp 5 . . . . 5 (℩𝑏𝐴𝑅𝑏) / 𝑏𝐴 = 𝐴
92csbvargi 4383 . . . . 5 (℩𝑏𝐴𝑅𝑏) / 𝑏𝑏 = (℩𝑏𝐴𝑅𝑏)
108, 9breq12i 5074 . . . 4 ((℩𝑏𝐴𝑅𝑏) / 𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏) / 𝑏𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏))
116, 10sylbb 221 . . 3 ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏))
124, 11syl 17 . 2 (∃!𝑏 𝐴𝑅𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏))
13 breldmg 5777 . 2 ((𝐴 ∈ V ∧ (℩𝑏𝐴𝑅𝑏) ∈ V ∧ 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) → 𝐴 ∈ dom 𝑅)
141, 3, 12, 13syl3anc 1367 1 (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1533  wcel 2110  ∃!weu 2649  Vcvv 3494  [wsbc 3771  csb 3882   class class class wbr 5065  dom cdm 5554  cio 6311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209  ax-pow 5265
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-dm 5564  df-iota 6313
This theorem is referenced by:  dfafv2  43330
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