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Theorem idsset 35872
Description: I is equal to the intersection of SSet and its converse. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
idsset I = ( SSet SSet )

Proof of Theorem idsset
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reli 5839 . 2 Rel I
2 relsset 35870 . . 3 Rel SSet
3 relin1 5825 . . 3 (Rel SSet → Rel ( SSet SSet ))
42, 3ax-mp 5 . 2 Rel ( SSet SSet )
5 eqss 4011 . . 3 (𝑦 = 𝑧 ↔ (𝑦𝑧𝑧𝑦))
6 vex 3482 . . . 4 𝑧 ∈ V
76ideq 5866 . . 3 (𝑦 I 𝑧𝑦 = 𝑧)
8 brin 5200 . . . 4 (𝑦( SSet SSet )𝑧 ↔ (𝑦 SSet 𝑧𝑦 SSet 𝑧))
96brsset 35871 . . . . 5 (𝑦 SSet 𝑧𝑦𝑧)
10 vex 3482 . . . . . . 7 𝑦 ∈ V
1110, 6brcnv 5896 . . . . . 6 (𝑦 SSet 𝑧𝑧 SSet 𝑦)
1210brsset 35871 . . . . . 6 (𝑧 SSet 𝑦𝑧𝑦)
1311, 12bitri 275 . . . . 5 (𝑦 SSet 𝑧𝑧𝑦)
149, 13anbi12i 628 . . . 4 ((𝑦 SSet 𝑧𝑦 SSet 𝑧) ↔ (𝑦𝑧𝑧𝑦))
158, 14bitri 275 . . 3 (𝑦( SSet SSet )𝑧 ↔ (𝑦𝑧𝑧𝑦))
165, 7, 153bitr4i 303 . 2 (𝑦 I 𝑧𝑦( SSet SSet )𝑧)
171, 4, 16eqbrriv 5804 1 I = ( SSet SSet )
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  cin 3962  wss 3963   class class class wbr 5148   I cid 5582  ccnv 5688  Rel wrel 5694   SSet csset 35814
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  ax-un 7754
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-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  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-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-sset 35838
This theorem is referenced by: (None)
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