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Theorem indislem 21175
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 6502 . . 3 (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
21preq2d 4493 . 2 (𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
3 dfsn2 4410 . . . 4 {∅} = {∅, ∅}
43eqcomi 2834 . . 3 {∅, ∅} = {∅}
5 fvprc 6426 . . . 4 𝐴 ∈ V → ( I ‘𝐴) = ∅)
65preq2d 4493 . . 3 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, ∅})
7 prprc2 4519 . . 3 𝐴 ∈ V → {∅, 𝐴} = {∅})
84, 6, 73eqtr4a 2887 . 2 𝐴 ∈ V → {∅, ( I ‘𝐴)} = {∅, 𝐴})
92, 8pm2.61i 177 1 {∅, ( I ‘𝐴)} = {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1656  wcel 2164  Vcvv 3414  c0 4144  {csn 4397  {cpr 4399   I cid 5249  cfv 6123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131
This theorem is referenced by:  indistop  21177  indisuni  21178  indiscld  21266  indisconn  21592  txindis  21808  hmphindis  21971
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