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Theorem moriotass 6293
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 3527 . . . . 5  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )
21imp 431 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph )  ->  E. x  e.  B  ph )
323adant3 1026 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B  ph )  ->  E. x  e.  B  ph )
4 simp3 1008 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B  ph )  ->  E* x  e.  B  ph )
5 reu5 3045 . . 3  |-  ( E! x  e.  B  ph  <->  ( E. x  e.  B  ph 
/\  E* x  e.  B  ph ) )
63, 4, 5sylanbrc 669 . 2  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B  ph )  ->  E! x  e.  B  ph )
7 riotass 6292 . 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 1310 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    /\ w3a 983    = wceq 1438   E.wrex 2777   E!wreu 2778   E*wrmo 2779    C_ wss 3437   iota_crio 6264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-un 3442  df-in 3444  df-ss 3451  df-sn 3998  df-pr 4000  df-uni 4218  df-iota 5563  df-riota 6265
This theorem is referenced by: (None)
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