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Theorem riotass 6294
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 3760 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
2 riotasbc 6282 . . . 4  |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A  ph )  /  x ]. ph )
31, 2syl 17 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  [. ( iota_ x  e.  A  ph )  /  x ]. ph )
4 simp1 1005 . . . . 5  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  A  C_  B
)
5 riotacl 6281 . . . . . 6  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
61, 5syl 17 . . . . 5  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A  ph )  e.  A )
74, 6sseldd 3471 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A  ph )  e.  B )
8 simp3 1007 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  B  ph )
9 nfriota1 6274 . . . . 5  |-  F/_ x
( iota_ x  e.  A  ph )
109nfsbc1 3324 . . . . 5  |-  F/ x [. ( iota_ x  e.  A  ph )  /  x ]. ph
11 sbceq1a 3316 . . . . 5  |-  ( x  =  ( iota_ x  e.  A  ph )  -> 
( ph  <->  [. ( iota_ x  e.  A  ph )  /  x ]. ph ) )
129, 10, 11riota2f 6288 . . . 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 665 . . 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 213 . 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 2437 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    = wceq 1437    e. wcel 1870   E.wrex 2783   E!wreu 2784   [.wsbc 3305    C_ wss 3442   iota_crio 6266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-un 3447  df-in 3449  df-ss 3456  df-sn 4003  df-pr 4005  df-uni 4223  df-iota 5565  df-riota 6267
This theorem is referenced by:  moriotass  6295
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