Theorem eusv4 5375

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 5369	. 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 3086	1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∃!weu 2596  ∀wral 3080  ∃wrex 3090  Vcvv 3454

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-nul 5266  ax-pow 5334

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-v 3456  df-dif 4050  df-nul 4282

This theorem is used by: (None)