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Theorem fvnonrel 42924
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 6915 . . 3 ((𝐴𝐴)‘𝑋) ∈ (ran (𝐴𝐴) ∪ {∅})
2 rnnonrel 42918 . . . . 5 ran (𝐴𝐴) = ∅
3 0ss 4391 . . . . 5 ∅ ⊆ {∅}
42, 3eqsstri 4011 . . . 4 ran (𝐴𝐴) ⊆ {∅}
5 ssequn1 4175 . . . 4 (ran (𝐴𝐴) ⊆ {∅} ↔ (ran (𝐴𝐴) ∪ {∅}) = {∅})
64, 5mpbi 229 . . 3 (ran (𝐴𝐴) ∪ {∅}) = {∅}
71, 6eleqtri 2825 . 2 ((𝐴𝐴)‘𝑋) ∈ {∅}
8 fvex 6898 . . 3 ((𝐴𝐴)‘𝑋) ∈ V
98elsn 4638 . 2 (((𝐴𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴𝐴)‘𝑋) = ∅)
107, 9mpbi 229 1 ((𝐴𝐴)‘𝑋) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  cdif 3940  cun 3941  wss 3943  c0 4317  {csn 4623  ccnv 5668  ran crn 5670  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-dm 5679  df-rn 5680  df-iota 6489  df-fv 6545
This theorem is referenced by: (None)
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