Theorem fvprcALT 6835

Description: Alternate proof of fvprc 6834 using ax-pow 5312 instead of ax-pr 5379. (Contributed by NM, 20-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
fvprcALT	⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)

Proof of Theorem fvprcALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brprcneuALT 6833	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
2		tz6.12-2 6829	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
3	1, 2	syl 17	1 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  ∃!weu 2569  Vcvv 3442  ∅c0 4287   class class class wbr 5100  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253  ax-pow 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508

This theorem is referenced by: (None)