Theorem eubrdm 47801

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 47800	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)
2		iotaex 6512	. . 3 ⊢ (℩𝑏𝐴𝑅𝑏) ∈ V
3	2	a1i 11	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → (℩𝑏𝐴𝑅𝑏) ∈ V)
4		iota4 6517	. . 3 ⊢ (∃!𝑏 𝐴𝑅𝑏 → [(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏)
5		sbcbr12g 5166	. . . . 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 3871	. . . . . 6 ⊢ ((℩𝑏𝐴𝑅𝑏) ∈ V → ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴)
8	2, 7	ax-mp 5	. . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴
9	2	csbvargi 4399	. . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 = (℩𝑏𝐴𝑅𝑏)
10	8, 9	breq12i 5117	. . . 4 ⊢ (⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 ↔ 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
11	6, 10	sylbb 222	. . 3 ⊢ ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
12	4, 11	syl 18	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
13		breldmg 5898	. 2 ⊢ ((𝐴 ∈ V ∧ (℩𝑏𝐴𝑅𝑏) ∈ V ∧ 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) → 𝐴 ∈ dom 𝑅)
14	1, 3, 12, 13	syl3anc 1397	1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2142  ∃!weu 2595  Vcvv 3454  [wsbc 3743  ⦋csb 3852   class class class wbr 5108  dom cdm 5660  ℩cio 6490

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-dm 5670  df-iota 6492

This theorem is used by:  dfafv2  47897