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Theorem fvnonrel 40698
 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 6690 . . 3 ((𝐴𝐴)‘𝑋) ∈ (ran (𝐴𝐴) ∪ {∅})
2 rnnonrel 40692 . . . . 5 ran (𝐴𝐴) = ∅
3 0ss 4295 . . . . 5 ∅ ⊆ {∅}
42, 3eqsstri 3928 . . . 4 ran (𝐴𝐴) ⊆ {∅}
5 ssequn1 4087 . . . 4 (ran (𝐴𝐴) ⊆ {∅} ↔ (ran (𝐴𝐴) ∪ {∅}) = {∅})
64, 5mpbi 233 . . 3 (ran (𝐴𝐴) ∪ {∅}) = {∅}
71, 6eleqtri 2850 . 2 ((𝐴𝐴)‘𝑋) ∈ {∅}
8 fvex 6675 . . 3 ((𝐴𝐴)‘𝑋) ∈ V
98elsn 4540 . 2 (((𝐴𝐴)‘𝑋) ∈ {∅} ↔ ((𝐴𝐴)‘𝑋) = ∅)
107, 9mpbi 233 1 ((𝐴𝐴)‘𝑋) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111   ∖ cdif 3857   ∪ cun 3858   ⊆ wss 3860  ∅c0 4227  {csn 4525  ◡ccnv 5526  ran crn 5528  ‘cfv 6339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-xp 5533  df-rel 5534  df-cnv 5535  df-dm 5537  df-rn 5538  df-iota 6298  df-fv 6347 This theorem is referenced by: (None)
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