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Theorem riotass 6192
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 3742 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
2 riotasbc 6180 . . . 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 988 . . . . 5  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  A  C_  B
)
5 riotacl 6179 . . . . . 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 3468 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A  ph )  e.  B )
8 simp3 990 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  B  ph )
9 nfriota1 6171 . . . . 5  |-  F/_ x
( iota_ x  e.  A  ph )
109nfsbc1 3313 . . . . 5  |-  F/ x [. ( iota_ x  e.  A  ph )  /  x ]. ph
11 sbceq1a 3305 . . . . 5  |-  ( x  =  ( iota_ x  e.  A  ph )  -> 
( ph  <->  [. ( iota_ x  e.  A  ph )  /  x ]. ph ) )
129, 10, 11riota2f 6186 . . . 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 661 . . 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 210 . 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 2462 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 184    /\ w3a 965    = wceq 1370    e. wcel 1758   E.wrex 2800   E!wreu 2801   [.wsbc 3294    C_ wss 3439   iota_crio 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-un 3444  df-in 3446  df-ss 3453  df-sn 3989  df-pr 3991  df-uni 4203  df-iota 5492  df-riota 6164
This theorem is referenced by:  moriotass  6193
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