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Theorem fvnonrel 41876
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 6873 . . 3 ((𝐴𝐴)‘𝑋) ∈ (ran (𝐴𝐴) ∪ {∅})
2 rnnonrel 41870 . . . . 5 ran (𝐴𝐴) = ∅
3 0ss 4357 . . . . 5 ∅ ⊆ {∅}
42, 3eqsstri 3979 . . . 4 ran (𝐴𝐴) ⊆ {∅}
5 ssequn1 4141 . . . 4 (ran (𝐴𝐴) ⊆ {∅} ↔ (ran (𝐴𝐴) ∪ {∅}) = {∅})
64, 5mpbi 229 . . 3 (ran (𝐴𝐴) ∪ {∅}) = {∅}
71, 6eleqtri 2836 . 2 ((𝐴𝐴)‘𝑋) ∈ {∅}
8 fvex 6856 . . 3 ((𝐴𝐴)‘𝑋) ∈ V
98elsn 4602 . 2 (((𝐴𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴𝐴)‘𝑋) = ∅)
107, 9mpbi 229 1 ((𝐴𝐴)‘𝑋) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  cdif 3908  cun 3909  wss 3911  c0 4283  {csn 4587  ccnv 5633  ran crn 5635  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-iota 6449  df-fv 6505
This theorem is referenced by: (None)
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