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Theorem riotass2 5437
Description: Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)
Assertion
Ref Expression
riotass2  C_  iota_  iota_
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem riotass2
StepHypRef Expression
1 reuss2 3211 . . . 4  C_
2 simplr 482 . . . 4  C_
3 riotasbc 5426 . . . . 5  [. iota_  ].
4 riotacl 5425 . . . . . 6  iota_
5 rspsbc 2834 . . . . . . 7 
iota_  [. iota_  ].
6 sbcimg 2798 . . . . . . 7 
iota_  [. iota_  ].  [. iota_  ].  [. iota_  ].
75, 6sylibd 138 . . . . . 6 
iota_  [. iota_  ].  [. iota_  ].
84, 7syl 14 . . . . 5  [. iota_  ].  [. iota_  ].
93, 8mpid 37 . . . 4  [. iota_  ].
101, 2, 9sylc 56 . . 3  C_  [. iota_  ].
111, 4syl 14 . . . . 5  C_  iota_
12 ssel 2933 . . . . . 6 
C_  iota_  iota_
1312ad2antrr 457 . . . . 5  C_  iota_  iota_
1411, 13mpd 13 . . . 4  C_  iota_
15 simprr 484 . . . 4  C_
16 nfriota1 5418 . . . . 5  F/_ iota_
1716nfsbc1 2775 . . . . 5  F/ [. iota_  ].
18 sbceq1a 2767 . . . . 5  iota_  [. iota_  ].
1916, 17, 18riota2f 5432 . . . 4  iota_  [. iota_  ].  iota_  iota_
2014, 15, 19syl2anc 391 . . 3  C_  [. iota_  ].  iota_  iota_
2110, 20mpbid 135 . 2  C_  iota_  iota_
2221eqcomd 2042 1  C_  iota_  iota_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wral 2300  wrex 2301  wreu 2302   [.wsbc 2758    C_ wss 2911   iota_crio 5410
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-riota 5411
This theorem is referenced by: (None)
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