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Theorem tz7.44lem1 6418
Description:  G is a function. Lemma for tz7.44-1 6419, tz7.44-2 6420, and tz7.44-3 6421. (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 5287 . . 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 5539 . . . 4  |-  ( H `
 ( x `  U. dom  x ) )  e.  _V
3 vex 2791 . . . . 5  |-  x  e. 
_V
4 rnexg 4940 . . . . 5  |-  ( x  e.  _V  ->  ran  x  e.  _V )
5 uniexg 4517 . . . . 5  |-  ( ran  x  e.  _V  ->  U.
ran  x  e.  _V )
63, 4, 5mp2b 9 . . . 4  |-  U. ran  x  e.  _V
7 nlim0 4450 . . . . . 6  |-  -.  Lim  (/)
8 dm0 4892 . . . . . . 7  |-  dom  (/)  =  (/)
9 limeq 4404 . . . . . . 7  |-  ( dom  (/)  =  (/)  ->  ( Lim 
dom  (/)  <->  Lim  (/) ) )
108, 9ax-mp 8 . . . . . 6  |-  ( Lim 
dom  (/)  <->  Lim  (/) )
117, 10mtbir 290 . . . . 5  |-  -.  Lim  dom  (/)
12 dmeq 4879 . . . . . . 7  |-  ( x  =  (/)  ->  dom  x  =  dom  (/) )
13 limeq 4404 . . . . . . 7  |-  ( dom  x  =  dom  (/)  ->  ( Lim  dom  x  <->  Lim  dom  (/) ) )
1412, 13syl 15 . . . . . 6  |-  ( x  =  (/)  ->  ( Lim 
dom  x  <->  Lim  dom  (/) ) )
1514biimpa 470 . . . . 5  |-  ( ( x  =  (/)  /\  Lim  dom  x )  ->  Lim  dom  (/) )
1611, 15mto 167 . . . 4  |-  -.  (
x  =  (/)  /\  Lim  dom  x )
172, 6, 16moeq3 2942 . . 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 1537 . 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 5275 . 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 200 1  |-  Fun  G
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   E*wmo 2144   _Vcvv 2788   (/)c0 3455   U.cuni 3827   {copab 4076   Lim wlim 4393   dom cdm 4689   ran crn 4690   Fun wfun 5249   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-lim 4397  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263
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