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Theorem fvnonrel 37722
 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 6203 . . 3 ((𝐴𝐴)‘𝑋) ∈ (ran (𝐴𝐴) ∪ {∅})
2 rnnonrel 37716 . . . . 5 ran (𝐴𝐴) = ∅
3 0ss 3963 . . . . 5 ∅ ⊆ {∅}
42, 3eqsstri 3627 . . . 4 ran (𝐴𝐴) ⊆ {∅}
5 ssequn1 3775 . . . 4 (ran (𝐴𝐴) ⊆ {∅} ↔ (ran (𝐴𝐴) ∪ {∅}) = {∅})
64, 5mpbi 220 . . 3 (ran (𝐴𝐴) ∪ {∅}) = {∅}
71, 6eleqtri 2697 . 2 ((𝐴𝐴)‘𝑋) ∈ {∅}
8 fvex 6188 . . 3 ((𝐴𝐴)‘𝑋) ∈ V
98elsn 4183 . 2 (((𝐴𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴𝐴)‘𝑋) = ∅)
107, 9mpbi 220 1 ((𝐴𝐴)‘𝑋) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1481   ∈ wcel 1988   ∖ cdif 3564   ∪ cun 3565   ⊆ wss 3567  ∅c0 3907  {csn 4168  ◡ccnv 5103  ran crn 5105  ‘cfv 5876 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-dm 5114  df-rn 5115  df-iota 5839  df-fv 5884 This theorem is referenced by: (None)
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