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Theorem idsset 35891
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 5836 . 2 Rel I
2 relsset 35889 . . 3 Rel SSet
3 relin1 5822 . . 3 (Rel SSet → Rel ( SSet SSet ))
42, 3ax-mp 5 . 2 Rel ( SSet SSet )
5 eqss 3999 . . 3 (𝑦 = 𝑧 ↔ (𝑦𝑧𝑧𝑦))
6 vex 3484 . . . 4 𝑧 ∈ V
76ideq 5863 . . 3 (𝑦 I 𝑧𝑦 = 𝑧)
8 brin 5195 . . . 4 (𝑦( SSet SSet )𝑧 ↔ (𝑦 SSet 𝑧𝑦 SSet 𝑧))
96brsset 35890 . . . . 5 (𝑦 SSet 𝑧𝑦𝑧)
10 vex 3484 . . . . . . 7 𝑦 ∈ V
1110, 6brcnv 5893 . . . . . 6 (𝑦 SSet 𝑧𝑧 SSet 𝑦)
1210brsset 35890 . . . . . 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 5801 1 I = ( SSet SSet )
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  cin 3950  wss 3951   class class class wbr 5143   I cid 5577  ccnv 5684  Rel wrel 5690   SSet csset 35833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-sset 35857
This theorem is referenced by: (None)
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