Theorem eubrdm 46953

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 46952	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ V)
2		iotaex 6548	. . 3 ⊢ (℩𝑏𝐴𝑅𝑏) ∈ V
3	2	a1i 11	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → (℩𝑏𝐴𝑅𝑏) ∈ V)
4		iota4 6556	. . 3 ⊢ (∃!𝑏 𝐴𝑅𝑏 → [(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏)
5		sbcbr12g 5222	. . . . 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 3940	. . . . . 6 ⊢ ((℩𝑏𝐴𝑅𝑏) ∈ V → ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴)
8	2, 7	ax-mp 5	. . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴 = 𝐴
9	2	csbvargi 4458	. . . . 5 ⊢ ⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 = (℩𝑏𝐴𝑅𝑏)
10	8, 9	breq12i 5175	. . . 4 ⊢ (⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝐴𝑅⦋(℩𝑏𝐴𝑅𝑏) / 𝑏⦌𝑏 ↔ 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
11	6, 10	sylbb 219	. . 3 ⊢ ([(℩𝑏𝐴𝑅𝑏) / 𝑏]𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
12	4, 11	syl 17	. 2 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴𝑅(℩𝑏𝐴𝑅𝑏))
13		breldmg 5934	. 2 ⊢ ((𝐴 ∈ V ∧ (℩𝑏𝐴𝑅𝑏) ∈ V ∧ 𝐴𝑅(℩𝑏𝐴𝑅𝑏)) → 𝐴 ∈ dom 𝑅)
14	1, 3, 12, 13	syl3anc 1371	1 ⊢ (∃!𝑏 𝐴𝑅𝑏 → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ∃!weu 2571  Vcvv 3488  [wsbc 3804  ⦋csb 3921   class class class wbr 5166  dom cdm 5700  ℩cio 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-dm 5710  df-iota 6527

This theorem is referenced by:  dfafv2  47049