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Theorem th3qcor 6782
Description: Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.)
Hypotheses
Ref Expression
th3q.1  |-  .~  e.  _V
th3q.2  |-  .~  Er  ( S  X.  S
)
th3q.4  |-  ( ( ( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  /\  ( (
s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S
) ) )  -> 
( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v
>.  .+  <. s ,  f
>. )  .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )
th3q.5  |-  G  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S
) /.  .~  )
)  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) }
Assertion
Ref Expression
th3qcor  |-  Fun  G
Distinct variable groups:    x, y,
z, w, v, u, t, s, f, g, h,  .~    x, S, y, z, w, v, u, t, s, f, g, h    x,  .+ , y, z, w, v, u, t, s, f, g, h
Allowed substitution hints:    G( x, y, z, w, v, u, t, f, g, h, s)

Proof of Theorem th3qcor
StepHypRef Expression
1 th3q.1 . . . . 5  |-  .~  e.  _V
2 th3q.2 . . . . 5  |-  .~  Er  ( S  X.  S
)
3 th3q.4 . . . . 5  |-  ( ( ( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  /\  ( (
s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S
) ) )  -> 
( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v
>.  .+  <. s ,  f
>. )  .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )
41, 2, 3th3qlem2 6781 . . . 4  |-  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S ) /.  .~  ) )  ->  E* z E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )
)
5 moanimv 2214 . . . 4  |-  ( E* z ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S ) /.  .~  ) )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
)  <->  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S ) /.  .~  ) )  ->  E* z E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )
) )
64, 5mpbir 200 . . 3  |-  E* z
( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S
) /.  .~  )
)  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
)
76funoprab 5960 . 2  |-  Fun  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S
) /.  .~  )
)  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) }
8 th3q.5 . . 3  |-  G  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S
) /.  .~  )
)  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) }
98funeqi 5291 . 2  |-  ( Fun 
G  <->  Fun  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  ( ( S  X.  S ) /.  .~  )  /\  y  e.  ( ( S  X.  S ) /.  .~  ) )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  .~  /\  y  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
) } )
107, 9mpbir 200 1  |-  Fun  G
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E*wmo 2157   _Vcvv 2801   <.cop 3656   class class class wbr 4039    X. cxp 4703   Fun wfun 5265  (class class class)co 5874   {coprab 5875    Er wer 6673   [cec 6674   /.cqs 6675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-er 6676  df-ec 6678  df-qs 6682
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