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Theorem th3qcor 6734
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 6733 . . . 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 2176 . . . 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 202 . . 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 5878 . 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 5214 . 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 202 1  |-  Fun  G
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   E*wmo 2119   _Vcvv 2763   <.cop 3617   class class class wbr 3997    X. cxp 4659   Fun wfun 4667  (class class class)co 5792   {coprab 5793    Er wer 6625   [cec 6626   /.cqs 6627
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-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  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-uni 3802  df-br 3998  df-opab 4052  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fv 4689  df-ov 5795  df-oprab 5796  df-er 6628  df-ec 6630  df-qs 6634
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