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Theorem idsset 31674
Description: I is equal to 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 5214 . 2 Rel I
2 relsset 31672 . . 3 Rel SSet
3 relin1 5202 . . 3 (Rel SSet → Rel ( SSet SSet ))
42, 3ax-mp 5 . 2 Rel ( SSet SSet )
5 eqss 3602 . . 3 (𝑦 = 𝑧 ↔ (𝑦𝑧𝑧𝑦))
6 vex 3192 . . . 4 𝑧 ∈ V
76ideq 5239 . . 3 (𝑦 I 𝑧𝑦 = 𝑧)
8 brin 4669 . . . 4 (𝑦( SSet SSet )𝑧 ↔ (𝑦 SSet 𝑧𝑦 SSet 𝑧))
96brsset 31673 . . . . 5 (𝑦 SSet 𝑧𝑦𝑧)
10 vex 3192 . . . . . . 7 𝑦 ∈ V
1110, 6brcnv 5270 . . . . . 6 (𝑦 SSet 𝑧𝑧 SSet 𝑦)
1210brsset 31673 . . . . . 6 (𝑧 SSet 𝑦𝑧𝑦)
1311, 12bitri 264 . . . . 5 (𝑦 SSet 𝑧𝑧𝑦)
149, 13anbi12i 732 . . . 4 ((𝑦 SSet 𝑧𝑦 SSet 𝑧) ↔ (𝑦𝑧𝑧𝑦))
158, 14bitri 264 . . 3 (𝑦( SSet SSet )𝑧 ↔ (𝑦𝑧𝑧𝑦))
165, 7, 153bitr4i 292 . 2 (𝑦 I 𝑧𝑦( SSet SSet )𝑧)
171, 4, 16eqbrriv 5181 1 I = ( SSet SSet )
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  cin 3558  wss 3559   class class class wbr 4618   I cid 4989  ccnv 5078  Rel wrel 5084   SSet csset 31615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31637  df-sset 31639
This theorem is referenced by: (None)
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