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Theorem ssoprab2 6122
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2 4472. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
ssoprab2  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  { <. <. x ,  y
>. ,  z >.  | 
ph }  C_  { <. <.
x ,  y >. ,  z >.  |  ps } )

Proof of Theorem ssoprab2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 id 20 . . . . . . . . . 10  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
21anim2d 549 . . . . . . . . 9  |-  ( (
ph  ->  ps )  -> 
( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
32alimi 1568 . . . . . . . 8  |-  ( A. z ( ph  ->  ps )  ->  A. z
( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
4 exim 1584 . . . . . . . 8  |-  ( A. z ( ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  (
w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) )  ->  ( E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph )  ->  E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ps ) ) )
53, 4syl 16 . . . . . . 7  |-  ( A. z ( ph  ->  ps )  ->  ( E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ps ) ) )
65alimi 1568 . . . . . 6  |-  ( A. y A. z ( ph  ->  ps )  ->  A. y
( E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) ) )
7 exim 1584 . . . . . 6  |-  ( A. y ( E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph )  ->  E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ps ) )  -> 
( E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) ) )
86, 7syl 16 . . . . 5  |-  ( A. y A. z ( ph  ->  ps )  ->  ( E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
98alimi 1568 . . . 4  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  A. x ( E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
10 exim 1584 . . . 4  |-  ( A. x ( E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) )  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
119, 10syl 16 . . 3  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  ( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ps ) ) )
1211ss2abdv 3408 . 2  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  { w  |  E. x E. y E. z
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) }  C_  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) } )
13 df-oprab 6077 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
14 df-oprab 6077 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ps }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ps ) }
1512, 13, 143sstr4g 3381 1  |-  ( A. x A. y A. z
( ph  ->  ps )  ->  { <. <. x ,  y
>. ,  z >.  | 
ph }  C_  { <. <.
x ,  y >. ,  z >.  |  ps } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652   {cab 2421    C_ wss 3312   <.cop 3809   {coprab 6074
This theorem is referenced by:  ssoprab2b  6123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-in 3319  df-ss 3326  df-oprab 6077
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