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Theorem moriotass 6546
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 3376 . . . . 5  |-  ( A 
C_  B  ->  ( E. x  e.  A  ph 
->  E. x  e.  B  ph ) )
21imp 419 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph )  ->  E. x  e.  B  ph )
323adant3 977 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B ph )  ->  E. x  e.  B  ph )
4 simp3 959 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B ph )  ->  E* x  e.  B ph )
5 reu5 2889 . . 3  |-  ( E! x  e.  B  ph  <->  ( E. x  e.  B  ph 
/\  E* x  e.  B ph ) )
63, 4, 5sylanbrc 646 . 2  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\ 
E* x  e.  B ph )  ->  E! x  e.  B  ph )
7 riotass 6545 . 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 1229 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 936    = wceq 1649   E.wrex 2675   E!wreu 2676   E*wrmo 2677    C_ wss 3288   iota_crio 6509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-iota 5385  df-fv 5429  df-riota 6516
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