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Theorem indislem 23023
Description: A lemma to eliminate some sethood hypotheses when dealing with the indiscrete topology. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
indislem {∅, ( I ‘𝐴)} = {∅, 𝐴}

Proof of Theorem indislem
StepHypRef Expression
1 fvi 6985 . . 3 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
21preq2d 4745 . 2 (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3 dfsn2 4644 . . . 4 {∅} = {∅, ∅}
43eqcomi 2744 . . 3 {∅, ∅} = {∅}
5 fvprc 6899 . . . 4 𝐴 ∈ V → ( I ‘𝐴) = ∅)
65preq2d 4745 . . 3 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7 prprc2 4771 . . 3 𝐴 ∈ V → {∅, 𝐴} = {∅})
84, 6, 73eqtr4a 2801 . 2 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
92, 8pm2.61i 182 1 {∅, ( I ‘𝐴)} = {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {csn 4631  {cpr 4633   I cid 5582  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  indistop  23025  indisuni  23026  indiscld  23115  indisconn  23442  txindis  23658  hmphindis  23821
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