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Theorem idsset 32600
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 5495 . 2 Rel I
2 relsset 32598 . . 3 Rel SSet
3 relin1 5483 . . 3 (Rel SSet → Rel ( SSet SSet ))
42, 3ax-mp 5 . 2 Rel ( SSet SSet )
5 eqss 3835 . . 3 (𝑦 = 𝑧 ↔ (𝑦𝑧𝑧𝑦))
6 vex 3400 . . . 4 𝑧 ∈ V
76ideq 5520 . . 3 (𝑦 I 𝑧𝑦 = 𝑧)
8 brin 4938 . . . 4 (𝑦( SSet SSet )𝑧 ↔ (𝑦 SSet 𝑧𝑦 SSet 𝑧))
96brsset 32599 . . . . 5 (𝑦 SSet 𝑧𝑦𝑧)
10 vex 3400 . . . . . . 7 𝑦 ∈ V
1110, 6brcnv 5550 . . . . . 6 (𝑦 SSet 𝑧𝑧 SSet 𝑦)
1210brsset 32599 . . . . . 6 (𝑧 SSet 𝑦𝑧𝑦)
1311, 12bitri 267 . . . . 5 (𝑦 SSet 𝑧𝑧𝑦)
149, 13anbi12i 620 . . . 4 ((𝑦 SSet 𝑧𝑦 SSet 𝑧) ↔ (𝑦𝑧𝑧𝑦))
158, 14bitri 267 . . 3 (𝑦( SSet SSet )𝑧 ↔ (𝑦𝑧𝑧𝑦))
165, 7, 153bitr4i 295 . 2 (𝑦 I 𝑧𝑦( SSet SSet )𝑧)
171, 4, 16eqbrriv 5462 1 I = ( SSet SSet )
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1601  cin 3790  wss 3791   class class class wbr 4886   I cid 5260  ccnv 5354  Rel wrel 5360   SSet csset 32542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-eprel 5266  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fo 6141  df-fv 6143  df-1st 7445  df-2nd 7446  df-txp 32564  df-sset 32566
This theorem is referenced by: (None)
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