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Theorem riotass 6349
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 3462 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
2 riotasbc 6336 . . . 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 6335 . . . . . 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 3194 . . . 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 6328 . . . . 5  |-  F/_ x
( iota_ x  e.  A ph )
109nfsbc1 3022 . . . . 5  |-  F/ x [. ( iota_ x  e.  A ph )  /  x ]. ph
11 sbceq1a 3014 . . . . 5  |-  ( x  =  ( iota_ x  e.  A ph )  -> 
( ph  <->  [. ( iota_ x  e.  A ph )  /  x ]. ph ) )
129, 10, 11riota2f 6342 . . . 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 2301 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 1632    e. wcel 1696   E.wrex 2557   E!wreu 2558   [.wsbc 3004    C_ wss 3165   iota_crio 6313
This theorem is referenced by:  moriotass  6350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-riota 6320
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