Theorem eusv4 5379

Description: Two ways to express single-valuedness of a class expression 𝐵(𝑦). (Contributed by NM, 27-Oct-2010.)

Hypothesis

Ref	Expression
eusv4.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
eusv4	⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem eusv4

Step	Hyp	Ref	Expression
1		reusv2lem3 5373	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝐵 ∈ V → (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵))
2		eusv4.1	. . 3 ⊢ 𝐵 ∈ V
3	2	a1i 11	. 2 ⊢ (𝑦 ∈ 𝐴 → 𝐵 ∈ V)
4	1, 3	mprg 3087	1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2146  ∃!weu 2598  ∀wral 3081  ∃wrex 3091  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-nul 5271  ax-pow 5338

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-v 3459  df-dif 3909  df-nul 4287

This theorem is used by: (None)