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 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅)

Distinct variable groups:   𝐴,𝑏   𝑅,𝑏

Proof of Theorem eubrdm

Step	Hyp	Ref	Expression
1		eubrv 46340	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)
2		iotaex 6515	. . 3 ⊢ (℩𝑏𝐴𝑅𝑏) ∈ V
3	2	a1i 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) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏))
6	2, 5	ax-mp 5	. . . 4 ⊢ ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 ↔ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏)
7		csbconstg 3908	. . . . . 6 ⊢ ((℩𝑏𝐴𝑅𝑏) ∈ V → ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴)
8	2, 7	ax-mp 5	. . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴
9	2	csbvargi 4428	. . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 = (℩𝑏𝐴𝑅𝑏)
10	8, 9	breq12i 5151	. . . 4 ⊢ (⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 ↔ 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
11	6, 10	sylbb 218	. . 3 ⊢ ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
12	4, 11	syl 17	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
13		breldmg 5906	. 2 ⊢ ((𝐴 ∈ V ∧ (℩𝑏𝐴𝑅𝑏) ∈ V ∧ 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) → 𝐴 ∈ dom 𝑅)
14	1, 3, 12, 13	syl3anc 1369	1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅)

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