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Theorem moriotass 6582
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 3410 . . . . 5  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )
21imp 420 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph )  ->  E. x  e.  B  ph )
323adant3 978 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B ph )  ->  E. x  e.  B  ph )
4 simp3 960 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B ph )  ->  E* x  e.  B ph )
5 reu5 2923 . . 3  |-  ( E! x  e.  B  ph  <->  ( E. x  e.  B  ph 
/\  E* x  e.  B ph ) )
63, 4, 5sylanbrc 647 . 2  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B ph )  ->  E! x  e.  B  ph )
7 riotass 6581 . 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 1230 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 937    = wceq 1653   E.wrex 2708   E!wreu 2709   E*wrmo 2710    C_ wss 3322   iota_crio 6545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-iota 5421  df-fv 5465  df-riota 6552
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