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Theorem riotass 6570
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 3614 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
2 riotasbc 6557 . . . 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 6556 . . . . . 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 3341 . . . 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 6549 . . . . 5  |-  F/_ x
( iota_ x  e.  A ph )
109nfsbc1 3171 . . . . 5  |-  F/ x [. ( iota_ x  e.  A ph )  /  x ]. ph
11 sbceq1a 3163 . . . . 5  |-  ( x  =  ( iota_ x  e.  A ph )  -> 
( ph  <->  [. ( iota_ x  e.  A ph )  /  x ]. ph ) )
129, 10, 11riota2f 6563 . . . 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 2440 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 1652    e. wcel 1725   E.wrex 2698   E!wreu 2699   [.wsbc 3153    C_ wss 3312   iota_crio 6534
This theorem is referenced by:  moriotass  6571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-riota 6541
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