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Theorem eubrdm 41961
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 41960 . 2 (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ V)
2 iotaex 6107 . . 3 (℩𝑏𝐴𝑅𝑏) ∈ V
32a1i 11 . 2 (∃!𝑏 𝐴𝑅𝑏 → (℩𝑏𝐴𝑅𝑏) ∈ V)
4 iota4 6108 . . 3 (∃!𝑏 𝐴𝑅𝑏[(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏)
5 sbcbr12g 4931 . . . . 5 ((℩𝑏𝐴𝑅𝑏) ∈ V → ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏(℩𝑏𝐴𝑅𝑏) / 𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏) / 𝑏𝑏))
62, 5ax-mp 5 . . . 4 ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏(℩𝑏𝐴𝑅𝑏) / 𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏) / 𝑏𝑏)
7 csbconstg 3770 . . . . . 6 ((℩𝑏𝐴𝑅𝑏) ∈ V → (℩𝑏𝐴𝑅𝑏) / 𝑏𝐴 = 𝐴)
82, 7ax-mp 5 . . . . 5 (℩𝑏𝐴𝑅𝑏) / 𝑏𝐴 = 𝐴
92csbvargi 4230 . . . . 5 (℩𝑏𝐴𝑅𝑏) / 𝑏𝑏 = (℩𝑏𝐴𝑅𝑏)
108, 9breq12i 4884 . . . 4 ((℩𝑏𝐴𝑅𝑏) / 𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏) / 𝑏𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏))
116, 10sylbb 211 . . 3 ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏))
124, 11syl 17 . 2 (∃!𝑏 𝐴𝑅𝑏𝐴𝑅(℩𝑏𝐴𝑅𝑏))
13 breldmg 5566 . 2 ((𝐴 ∈ V ∧ (℩𝑏𝐴𝑅𝑏) ∈ V ∧ 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) → 𝐴 ∈ dom 𝑅)
141, 3, 12, 13syl3anc 1494 1 (∃!𝑏 𝐴𝑅𝑏𝐴 ∈ dom 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1656  wcel 2164  ∃!weu 2639  Vcvv 3414  [wsbc 3662  csb 3757   class class class wbr 4875  dom cdm 5346  cio 6088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015  ax-pow 5067
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-dm 5356  df-iota 6090
This theorem is referenced by:  dfafv2  42028
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