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Theorem idsset 31167
 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 5171 . 2 Rel I
2 relsset 31165 . . 3 Rel SSet
3 relin1 5159 . . 3 (Rel SSet → Rel ( SSet SSet ))
42, 3ax-mp 5 . 2 Rel ( SSet SSet )
5 eqss 3583 . . 3 (𝑦 = 𝑧 ↔ (𝑦𝑧𝑧𝑦))
6 vex 3176 . . . 4 𝑧 ∈ V
76ideq 5196 . . 3 (𝑦 I 𝑧𝑦 = 𝑧)
8 brin 4634 . . . 4 (𝑦( SSet SSet )𝑧 ↔ (𝑦 SSet 𝑧𝑦 SSet 𝑧))
96brsset 31166 . . . . 5 (𝑦 SSet 𝑧𝑦𝑧)
10 vex 3176 . . . . . . 7 𝑦 ∈ V
1110, 6brcnv 5227 . . . . . 6 (𝑦 SSet 𝑧𝑧 SSet 𝑦)
1210brsset 31166 . . . . . 6 (𝑧 SSet 𝑦𝑧𝑦)
1311, 12bitri 263 . . . . 5 (𝑦 SSet 𝑧𝑧𝑦)
149, 13anbi12i 729 . . . 4 ((𝑦 SSet 𝑧𝑦 SSet 𝑧) ↔ (𝑦𝑧𝑧𝑦))
158, 14bitri 263 . . 3 (𝑦( SSet SSet )𝑧 ↔ (𝑦𝑧𝑧𝑦))
165, 7, 153bitr4i 291 . 2 (𝑦 I 𝑧𝑦( SSet SSet )𝑧)
171, 4, 16eqbrriv 5138 1 I = ( SSet SSet )
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∩ cin 3539   ⊆ wss 3540   class class class wbr 4583   I cid 4948  ◡ccnv 5037  Rel wrel 5043   SSet csset 31108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-sset 31132 This theorem is referenced by: (None)
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