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Theorem indislem 22858
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 6961 . . 3 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
21preq2d 4739 . 2 (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3 dfsn2 4636 . . . 4 {∅} = {∅, ∅}
43eqcomi 2735 . . 3 {∅, ∅} = {∅}
5 fvprc 6877 . . . 4 𝐴 ∈ V → ( I ‘𝐴) = ∅)
65preq2d 4739 . . 3 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7 prprc2 4765 . . 3 𝐴 ∈ V → {∅, 𝐴} = {∅})
84, 6, 73eqtr4a 2792 . 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 1533  wcel 2098  Vcvv 3468  c0 4317  {csn 4623  {cpr 4625   I cid 5566  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-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-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545
This theorem is referenced by:  indistop  22860  indisuni  22861  indiscld  22950  indisconn  23277  txindis  23493  hmphindis  23656
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