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Theorem moriotass 6421
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 3314 . . . . 5  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )
21imp 418 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph )  ->  E. x  e.  B  ph )
323adant3 975 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B ph )  ->  E. x  e.  B  ph )
4 simp3 957 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B ph )  ->  E* x  e.  B ph )
5 reu5 2829 . . 3  |-  ( E! x  e.  B  ph  <->  ( E. x  e.  B  ph 
/\  E* x  e.  B ph ) )
63, 4, 5sylanbrc 645 . 2  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B ph )  ->  E! x  e.  B  ph )
7 riotass 6420 . 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 1227 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 934    = wceq 1642   E.wrex 2620   E!wreu 2621   E*wrmo 2622    C_ wss 3228   iota_crio 6384
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345  df-riota 6391
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