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Theorem ssoprab2g 24398
Description: Inference of operation class abstraction subclass from implication. (Contributed by FL, 24-Jan-2010.)
Hypothesis
Ref Expression
ssoprab2g.1  |-  ( ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
ssoprab2g  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  C_  { <. <.
x ,  y >. ,  z >.  |  ch } )
Distinct variable groups:    ph, x, z    ph, y, z
Allowed substitution hints:    ps( x, y, z)    ch( x, y, z)

Proof of Theorem ssoprab2g
StepHypRef Expression
1 ssoprab2g.1 . . . . 5  |-  ( ph  ->  ( ps  ->  ch ) )
21anim2d 550 . . . 4  |-  ( ph  ->  ( ( u  = 
<. x ,  y >.  /\  ps )  ->  (
u  =  <. x ,  y >.  /\  ch ) ) )
322eximdv 2021 . . 3  |-  ( ph  ->  ( E. x E. y ( u  = 
<. x ,  y >.  /\  ps )  ->  E. x E. y ( u  = 
<. x ,  y >.  /\  ch ) ) )
43ssopab2dv 4265 . 2  |-  ( ph  ->  { <. u ,  z
>.  |  E. x E. y ( u  = 
<. x ,  y >.  /\  ps ) }  C_  {
<. u ,  z >.  |  E. x E. y
( u  =  <. x ,  y >.  /\  ch ) } )
5 dfoprab2 5829 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ps ) }
6 dfoprab2 5829 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ch }  =  { <. u ,  z >.  |  E. x E. y ( u  =  <. x ,  y
>.  /\  ch ) }
74, 5, 63sstr4g 3194 1  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  ps }  C_  { <. <.
x ,  y >. ,  z >.  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    C_ wss 3127   <.cop 3617   {copab 4050   {coprab 5793
This theorem is referenced by:  oprabex2gpop  24402
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-opab 4052  df-oprab 5796
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