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Theorem idsset 33378
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 5686 . 2 Rel I
2 relsset 33376 . . 3 Rel SSet
3 relin1 5673 . . 3 (Rel SSet → Rel ( SSet SSet ))
42, 3ax-mp 5 . 2 Rel ( SSet SSet )
5 eqss 3968 . . 3 (𝑦 = 𝑧 ↔ (𝑦𝑧𝑧𝑦))
6 vex 3483 . . . 4 𝑧 ∈ V
76ideq 5711 . . 3 (𝑦 I 𝑧𝑦 = 𝑧)
8 brin 5105 . . . 4 (𝑦( SSet SSet )𝑧 ↔ (𝑦 SSet 𝑧𝑦 SSet 𝑧))
96brsset 33377 . . . . 5 (𝑦 SSet 𝑧𝑦𝑧)
10 vex 3483 . . . . . . 7 𝑦 ∈ V
1110, 6brcnv 5741 . . . . . 6 (𝑦 SSet 𝑧𝑧 SSet 𝑦)
1210brsset 33377 . . . . . 6 (𝑧 SSet 𝑦𝑧𝑦)
1311, 12bitri 278 . . . . 5 (𝑦 SSet 𝑧𝑧𝑦)
149, 13anbi12i 629 . . . 4 ((𝑦 SSet 𝑧𝑦 SSet 𝑧) ↔ (𝑦𝑧𝑧𝑦))
158, 14bitri 278 . . 3 (𝑦( SSet SSet )𝑧 ↔ (𝑦𝑧𝑧𝑦))
165, 7, 153bitr4i 306 . 2 (𝑦 I 𝑧𝑦( SSet SSet )𝑧)
171, 4, 16eqbrriv 5652 1 I = ( SSet SSet )
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  cin 3918  wss 3919   class class class wbr 5053   I cid 5447  ccnv 5542  Rel wrel 5548   SSet csset 33320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-eprel 5453  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fo 6350  df-fv 6352  df-1st 7681  df-2nd 7682  df-txp 33342  df-sset 33344
This theorem is referenced by: (None)
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