Theorem rnfvprc 6657

Description: The range of a function value at a proper class is empty. (Contributed by AV, 20-Aug-2022.)

Hypothesis

Ref	Expression
rnfvprc.y	⊢ 𝑌 = (𝐹‘𝑋)

Assertion

Ref	Expression
rnfvprc	⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅)

Proof of Theorem rnfvprc

Step	Hyp	Ref	Expression
1		rnfvprc.y	. . . 4 ⊢ 𝑌 = (𝐹‘𝑋)
2		fvprc 6656	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
3	1, 2	syl5eq 2867	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅)
4	3	rneqd 5801	. 2 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ran ∅)
5		rn0 5789	. 2 ⊢ ran ∅ = ∅
6	4, 5	syl6eq 2871	1 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1536   ∈ wcel 2113  Vcvv 3491  ∅c0 4284  ran crn 5549  ‘cfv 6348

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-cnv 5556  df-dm 5558  df-rn 5559  df-iota 6307  df-fv 6356

This theorem is referenced by:  pmtrfrn  18579  mrsubrn  32779  mrsub0  32782  mrsubf  32783  mrsubccat  32784  mrsubcn  32785  mrsubco  32787  mrsubvrs  32788  elmsubrn  32794  msubrn  32795  msubf  32798