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Theorem moriotass 5751
Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
moriotass  |-  ( ( 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 moriotass
StepHypRef Expression
1 ssrexv 3157 . . . . 5  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )
21imp 123 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph )  ->  E. x  e.  B  ph )
323adant3 1001 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B  ph )  ->  E. x  e.  B  ph )
4 simp3 983 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B  ph )  ->  E* x  e.  B  ph )
5 reu5 2641 . . 3  |-  ( E! x  e.  B  ph  <->  ( E. x  e.  B  ph 
/\  E* x  e.  B  ph ) )
63, 4, 5sylanbrc 413 . 2  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B  ph )  ->  E! x  e.  B  ph )
7 riotass 5750 . 2  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A  ph )  =  ( iota_ x  e.  B  ph ) )
86, 7syld3an3 1261 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    /\ w3a 962    = wceq 1331   E.wrex 2415   E!wreu 2416   E*wrmo 2417    C_ wss 3066   iota_crio 5722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-uni 3732  df-iota 5083  df-riota 5723
This theorem is referenced by: (None)
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