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Metamath Proof Explorer

Theorem fvprc

Description: A function's value at a proper class is the empty set. See fvprcALT for a proof that uses ax-pow instead of ax-pr . (Contributed by NM, 20-May-1998) Avoid ax-pow . (Revised by BTernaryTau, 3-Aug-2024) (Proof shortened by BTernaryTau, 3-Dec-2024)

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

Proof

Step	Hyp	Ref	Expression
1		brprcneu	⊢ ( ¬ 𝐴 ∈ V → ¬ ∃! 𝑥 𝐴 𝐹 𝑥 )
2		tz6.12-2	⊢ ( ¬ ∃! 𝑥 𝐴 𝐹 𝑥 → ( 𝐹 ‘ 𝐴 ) = ∅ )
3	1 2	syl	⊢ ( ¬ 𝐴 ∈ V → ( 𝐹 ‘ 𝐴 ) = ∅ )