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Theorem idsset 32334
 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 5388 . 2 Rel I
2 relsset 32332 . . 3 Rel SSet
3 relin1 5375 . . 3 (Rel SSet → Rel ( SSet SSet ))
42, 3ax-mp 5 . 2 Rel ( SSet SSet )
5 eqss 3767 . . 3 (𝑦 = 𝑧 ↔ (𝑦𝑧𝑧𝑦))
6 vex 3354 . . . 4 𝑧 ∈ V
76ideq 5413 . . 3 (𝑦 I 𝑧𝑦 = 𝑧)
8 brin 4838 . . . 4 (𝑦( SSet SSet )𝑧 ↔ (𝑦 SSet 𝑧𝑦 SSet 𝑧))
96brsset 32333 . . . . 5 (𝑦 SSet 𝑧𝑦𝑧)
10 vex 3354 . . . . . . 7 𝑦 ∈ V
1110, 6brcnv 5443 . . . . . 6 (𝑦 SSet 𝑧𝑧 SSet 𝑦)
1210brsset 32333 . . . . . 6 (𝑧 SSet 𝑦𝑧𝑦)
1311, 12bitri 264 . . . . 5 (𝑦 SSet 𝑧𝑧𝑦)
149, 13anbi12i 612 . . . 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 5355 1 I = ( SSet SSet )
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 382   = wceq 1631   ∩ cin 3722   ⊆ wss 3723   class class class wbr 4786   I cid 5156  ◡ccnv 5248  Rel wrel 5254   SSet csset 32276 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-eprel 5162  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7315  df-2nd 7316  df-txp 32298  df-sset 32300 This theorem is referenced by: (None)
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