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Theorem riotass 6333
Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
riotass  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  B ph ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem riotass
StepHypRef Expression
1 reuss 3449 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
2 riotasbc 6320 . . . 4  |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A ph )  /  x ]. ph )
31, 2syl 15 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  [. ( iota_ x  e.  A ph )  /  x ]. ph )
4 simp1 955 . . . . 5  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  A  C_  B
)
5 riotacl 6319 . . . . . 6  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A
)
61, 5syl 15 . . . . 5  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A ph )  e.  A )
74, 6sseldd 3181 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A ph )  e.  B )
8 simp3 957 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  B  ph )
9 nfriota1 6312 . . . . 5  |-  F/_ x
( iota_ x  e.  A ph )
109nfsbc1 3009 . . . . 5  |-  F/ x [. ( iota_ x  e.  A ph )  /  x ]. ph
11 sbceq1a 3001 . . . . 5  |-  ( x  =  ( iota_ x  e.  A ph )  -> 
( ph  <->  [. ( iota_ x  e.  A ph )  /  x ]. ph ) )
129, 10, 11riota2f 6326 . . . 4  |-  ( ( ( iota_ x  e.  A ph )  e.  B  /\  E! x  e.  B  ph )  ->  ( [. ( iota_ x  e.  A ph )  /  x ]. ph  <->  ( iota_ x  e.  B ph )  =  ( iota_ x  e.  A ph ) ) )
137, 8, 12syl2anc 642 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( [. ( iota_ x  e.  A ph )  /  x ]. ph  <->  ( iota_ x  e.  B ph )  =  ( iota_ x  e.  A ph ) ) )
143, 13mpbid 201 . 2  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  B ph )  =  ( iota_ x  e.  A ph ) )
1514eqcomd 2288 1  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A ph )  =  ( iota_ x  e.  B ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   E!wreu 2545   [.wsbc 2991    C_ wss 3152   iota_crio 6297
This theorem is referenced by:  moriotass  6334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-riota 6304
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