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Theorem fvnonrel 39835
Description: The function value of any class under a non-relation is empty. (Contributed by RP, 23-Oct-2020.)
Assertion
Ref Expression
fvnonrel ((𝐴𝐴)‘𝑋) = ∅

Proof of Theorem fvnonrel
StepHypRef Expression
1 fvrn0 6691 . . 3 ((𝐴𝐴)‘𝑋) ∈ (ran (𝐴𝐴) ∪ {∅})
2 rnnonrel 39829 . . . . 5 ran (𝐴𝐴) = ∅
3 0ss 4347 . . . . 5 ∅ ⊆ {∅}
42, 3eqsstri 3998 . . . 4 ran (𝐴𝐴) ⊆ {∅}
5 ssequn1 4153 . . . 4 (ran (𝐴𝐴) ⊆ {∅} ↔ (ran (𝐴𝐴) ∪ {∅}) = {∅})
64, 5mpbi 231 . . 3 (ran (𝐴𝐴) ∪ {∅}) = {∅}
71, 6eleqtri 2908 . 2 ((𝐴𝐴)‘𝑋) ∈ {∅}
8 fvex 6676 . . 3 ((𝐴𝐴)‘𝑋) ∈ V
98elsn 4572 . 2 (((𝐴𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴𝐴)‘𝑋) = ∅)
107, 9mpbi 231 1 ((𝐴𝐴)‘𝑋) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  cdif 3930  cun 3931  wss 3933  c0 4288  {csn 4557  ccnv 5547  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 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-iota 6307  df-fv 6356
This theorem is referenced by: (None)
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