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Theorem tz7.44lem1 7063
Description:  G is a function. Lemma for tz7.44-1 7064, tz7.44-2 7065, and tz7.44-3 7066. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
tz7.44lem1.1  |-  G  =  { <. x ,  y
>.  |  ( (
x  =  (/)  /\  y  =  A )  \/  ( -.  ( x  =  (/)  \/ 
Lim  dom  x )  /\  y  =  ( H `  ( x `  U. dom  x ) ) )  \/  ( Lim  dom  x  /\  y  =  U. ran  x ) ) }
Assertion
Ref Expression
tz7.44lem1  |-  Fun  G
Distinct variable groups:    x, y    y, A    y, H
Allowed substitution hints:    A( x)    G( x, y)    H( x)

Proof of Theorem tz7.44lem1
StepHypRef Expression
1 funopab 5603 . . 3  |-  ( Fun 
{ <. x ,  y
>.  |  ( (
x  =  (/)  /\  y  =  A )  \/  ( -.  ( x  =  (/)  \/ 
Lim  dom  x )  /\  y  =  ( H `  ( x `  U. dom  x ) ) )  \/  ( Lim  dom  x  /\  y  =  U. ran  x ) ) }  <->  A. x E* y ( ( x  =  (/)  /\  y  =  A )  \/  ( -.  (
x  =  (/)  \/  Lim  dom  x )  /\  y  =  ( H `  ( x `  U. dom  x ) ) )  \/  ( Lim  dom  x  /\  y  =  U. ran  x ) ) )
2 fvex 5858 . . . 4  |-  ( H `
 ( x `  U. dom  x ) )  e.  _V
3 vex 3109 . . . . 5  |-  x  e. 
_V
4 rnexg 6705 . . . . 5  |-  ( x  e.  _V  ->  ran  x  e.  _V )
5 uniexg 6570 . . . . 5  |-  ( ran  x  e.  _V  ->  U.
ran  x  e.  _V )
63, 4, 5mp2b 10 . . . 4  |-  U. ran  x  e.  _V
7 nlim0 4925 . . . . . 6  |-  -.  Lim  (/)
8 dm0 5205 . . . . . . 7  |-  dom  (/)  =  (/)
9 limeq 4879 . . . . . . 7  |-  ( dom  (/)  =  (/)  ->  ( Lim 
dom  (/)  <->  Lim  (/) ) )
108, 9ax-mp 5 . . . . . 6  |-  ( Lim 
dom  (/)  <->  Lim  (/) )
117, 10mtbir 297 . . . . 5  |-  -.  Lim  dom  (/)
12 dmeq 5192 . . . . . . 7  |-  ( x  =  (/)  ->  dom  x  =  dom  (/) )
13 limeq 4879 . . . . . . 7  |-  ( dom  x  =  dom  (/)  ->  ( Lim  dom  x  <->  Lim  dom  (/) ) )
1412, 13syl 16 . . . . . 6  |-  ( x  =  (/)  ->  ( Lim 
dom  x  <->  Lim  dom  (/) ) )
1514biimpa 482 . . . . 5  |-  ( ( x  =  (/)  /\  Lim  dom  x )  ->  Lim  dom  (/) )
1611, 15mto 176 . . . 4  |-  -.  (
x  =  (/)  /\  Lim  dom  x )
172, 6, 16moeq3 3273 . . 3  |-  E* y
( ( x  =  (/)  /\  y  =  A )  \/  ( -.  ( x  =  (/)  \/ 
Lim  dom  x )  /\  y  =  ( H `  ( x `  U. dom  x ) ) )  \/  ( Lim  dom  x  /\  y  =  U. ran  x ) )
181, 17mpgbir 1627 . 2  |-  Fun  { <. x ,  y >.  |  ( ( x  =  (/)  /\  y  =  A )  \/  ( -.  ( x  =  (/)  \/ 
Lim  dom  x )  /\  y  =  ( H `  ( x `  U. dom  x ) ) )  \/  ( Lim  dom  x  /\  y  =  U. ran  x ) ) }
19 tz7.44lem1.1 . . 3  |-  G  =  { <. x ,  y
>.  |  ( (
x  =  (/)  /\  y  =  A )  \/  ( -.  ( x  =  (/)  \/ 
Lim  dom  x )  /\  y  =  ( H `  ( x `  U. dom  x ) ) )  \/  ( Lim  dom  x  /\  y  =  U. ran  x ) ) }
2019funeqi 5590 . 2  |-  ( Fun 
G  <->  Fun  { <. x ,  y >.  |  ( ( x  =  (/)  /\  y  =  A )  \/  ( -.  (
x  =  (/)  \/  Lim  dom  x )  /\  y  =  ( H `  ( x `  U. dom  x ) ) )  \/  ( Lim  dom  x  /\  y  =  U. ran  x ) ) } )
2118, 20mpbir 209 1  |-  Fun  G
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 366    /\ wa 367    \/ w3o 970    = wceq 1398    e. wcel 1823   E*wmo 2285   _Vcvv 3106   (/)c0 3783   U.cuni 4235   {copab 4496   Lim wlim 4868   dom cdm 4988   ran crn 4989   Fun wfun 5564   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-lim 4872  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578
This theorem is referenced by: (None)
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