Theorem eubrdm 47513

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 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅)

Distinct variable groups:   𝐴,𝑏   𝑅,𝑏

Proof of Theorem eubrdm

Step	Hyp	Ref	Expression
1		eubrv 47512	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)
2		iotaex 6465	. . 3 ⊢ (℩𝑏𝐴𝑅𝑏) ∈ V
3	2	a1i 11	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → (℩𝑏𝐴𝑅𝑏) ∈ V)
4		iota4 6470	. . 3 ⊢ (∃!𝑏 𝐴𝑅𝑏 → [(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏)
5		sbcbr12g 5131	. . . . 5 ⊢ ((℩𝑏𝐴𝑅𝑏) ∈ V → ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 ↔ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏))
6	2, 5	ax-mp 5	. . . 4 ⊢ ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 ↔ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏)
7		csbconstg 3852	. . . . . 6 ⊢ ((℩𝑏𝐴𝑅𝑏) ∈ V → ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴)
8	2, 7	ax-mp 5	. . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴
9	2	csbvargi 4366	. . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 = (℩𝑏𝐴𝑅𝑏)
10	8, 9	breq12i 5084	. . . 4 ⊢ (⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 ↔ 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
11	6, 10	sylbb 221	. . 3 ⊢ ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
12	4, 11	syl 17	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
13		breldmg 5858	. 2 ⊢ ((𝐴 ∈ V ∧ (℩𝑏𝐴𝑅𝑏) ∈ V ∧ 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) → 𝐴 ∈ dom 𝑅)
14	1, 3, 12, 13	syl3anc 1380	1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548   ∈ wcel 2121  ∃!weu 2574  Vcvv 3433  [wsbc 3725  ⦋csb 3833   class class class wbr 5075  dom cdm 5621  ℩cio 6443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-dm 5631  df-iota 6445

This theorem is referenced by:  dfafv2  47609