Theorem fvprcALT 6833

Description: Alternate proof of fvprc 6832 using ax-pow 5315 instead of ax-pr 5382. (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 6831	. 2 ⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
2		tz6.12-2 6828	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
3	1, 2	syl 17	1 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  ∃!weu 2561  Vcvv 3444  ∅c0 4292   class class class wbr 5102  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-nul 5256  ax-pow 5315

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507

This theorem is referenced by: (None)