Theorem rnfvprc 6426

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 6425	. . . 4 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅)
3	1, 2	syl5eq 2872	. . 3 ⊢ (¬ 𝑋 ∈ V → 𝑌 = ∅)
4	3	rneqd 5584	. 2 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ran ∅)
5		rn0 5609	. 2 ⊢ ran ∅ = ∅
6	4, 5	syl6eq 2876	1 ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1658   ∈ wcel 2166  Vcvv 3413  ∅c0 4143  ran crn 5342  ‘cfv 6122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-cnv 5349  df-dm 5351  df-rn 5352  df-iota 6085  df-fv 6130

This theorem is referenced by:  pmtrfrn  18227  mrsubrn  31955  mrsub0  31958  mrsubf  31959  mrsubccat  31960  mrsubcn  31961  mrsubco  31963  mrsubvrs  31964  elmsubrn  31970  msubrn  31971  msubf  31974