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Theorem idsset 35491
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 5830 . 2 Rel I
2 relsset 35489 . . 3 Rel SSet
3 relin1 5816 . . 3 (Rel SSet → Rel ( SSet SSet ))
42, 3ax-mp 5 . 2 Rel ( SSet SSet )
5 eqss 3995 . . 3 (𝑦 = 𝑧 ↔ (𝑦𝑧𝑧𝑦))
6 vex 3475 . . . 4 𝑧 ∈ V
76ideq 5857 . . 3 (𝑦 I 𝑧𝑦 = 𝑧)
8 brin 5202 . . . 4 (𝑦( SSet SSet )𝑧 ↔ (𝑦 SSet 𝑧𝑦 SSet 𝑧))
96brsset 35490 . . . . 5 (𝑦 SSet 𝑧𝑦𝑧)
10 vex 3475 . . . . . . 7 𝑦 ∈ V
1110, 6brcnv 5887 . . . . . 6 (𝑦 SSet 𝑧𝑧 SSet 𝑦)
1210brsset 35490 . . . . . 6 (𝑧 SSet 𝑦𝑧𝑦)
1311, 12bitri 274 . . . . 5 (𝑦 SSet 𝑧𝑧𝑦)
149, 13anbi12i 626 . . . 4 ((𝑦 SSet 𝑧𝑦 SSet 𝑧) ↔ (𝑦𝑧𝑧𝑦))
158, 14bitri 274 . . 3 (𝑦( SSet SSet )𝑧 ↔ (𝑦𝑧𝑧𝑦))
165, 7, 153bitr4i 302 . 2 (𝑦 I 𝑧𝑦( SSet SSet )𝑧)
171, 4, 16eqbrriv 5795 1 I = ( SSet SSet )
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  cin 3946  wss 3947   class class class wbr 5150   I cid 5577  ccnv 5679  Rel wrel 5685   SSet csset 35433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-eprel 5584  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fo 6557  df-fv 6559  df-1st 7997  df-2nd 7998  df-txp 35455  df-sset 35457
This theorem is referenced by: (None)
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