Theorem eusv4 5346

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 5340	. 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 3054	1 ⊢ (∃!𝑥∃𝑦 ∈ 𝐴 𝑥 = 𝐵 ↔ ∃!𝑥∀𝑦 ∈ 𝐴 𝑥 = 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ∃!weu 2565  ∀wral 3048  ∃wrex 3057  Vcvv 3437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-nul 5246  ax-pow 5305

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-v 3439  df-dif 3901  df-nul 4283

This theorem is referenced by: (None)