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Theorem riotass 6514
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 3565 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
2 riotasbc 6501 . . . 4  |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A ph )  /  x ]. ph )
31, 2syl 16 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  [. ( iota_ x  e.  A ph )  /  x ]. ph )
4 simp1 957 . . . . 5  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  A  C_  B
)
5 riotacl 6500 . . . . . 6  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A ph )  e.  A
)
61, 5syl 16 . . . . 5  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A ph )  e.  A )
74, 6sseldd 3292 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A ph )  e.  B )
8 simp3 959 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  B  ph )
9 nfriota1 6493 . . . . 5  |-  F/_ x
( iota_ x  e.  A ph )
109nfsbc1 3122 . . . . 5  |-  F/ x [. ( iota_ x  e.  A ph )  /  x ]. ph
11 sbceq1a 3114 . . . . 5  |-  ( x  =  ( iota_ x  e.  A ph )  -> 
( ph  <->  [. ( iota_ x  e.  A ph )  /  x ]. ph ) )
129, 10, 11riota2f 6507 . . . 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 643 . . 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 202 . 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 2392 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 177    /\ w3a 936    = wceq 1649    e. wcel 1717   E.wrex 2650   E!wreu 2651   [.wsbc 3104    C_ wss 3263   iota_crio 6478
This theorem is referenced by:  moriotass  6515
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402  df-riota 6485
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