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Theorem fvnonrel 43610
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 6936 . . 3 ((𝐴𝐴)‘𝑋) ∈ (ran (𝐴𝐴) ∪ {∅})
2 rnnonrel 43604 . . . . 5 ran (𝐴𝐴) = ∅
3 0ss 4400 . . . . 5 ∅ ⊆ {∅}
42, 3eqsstri 4030 . . . 4 ran (𝐴𝐴) ⊆ {∅}
5 ssequn1 4186 . . . 4 (ran (𝐴𝐴) ⊆ {∅} ↔ (ran (𝐴𝐴) ∪ {∅}) = {∅})
64, 5mpbi 230 . . 3 (ran (𝐴𝐴) ∪ {∅}) = {∅}
71, 6eleqtri 2839 . 2 ((𝐴𝐴)‘𝑋) ∈ {∅}
8 fvex 6919 . . 3 ((𝐴𝐴)‘𝑋) ∈ V
98elsn 4641 . 2 (((𝐴𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴𝐴)‘𝑋) = ∅)
107, 9mpbi 230 1 ((𝐴𝐴)‘𝑋) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  cdif 3948  cun 3949  wss 3951  c0 4333  {csn 4626  ccnv 5684  ran crn 5686  cfv 6561
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-iota 6514  df-fv 6569
This theorem is referenced by: (None)
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