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Theorem tz7.44lem1 6882
Description:  G is a function. Lemma for tz7.44-1 6883, tz7.44-2 6884, and tz7.44-3 6885. (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 5472 . . 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 5722 . . . 4  |-  ( H `
 ( x `  U. dom  x ) )  e.  _V
3 vex 2996 . . . . 5  |-  x  e. 
_V
4 rnexg 6531 . . . . 5  |-  ( x  e.  _V  ->  ran  x  e.  _V )
5 uniexg 6398 . . . . 5  |-  ( ran  x  e.  _V  ->  U.
ran  x  e.  _V )
63, 4, 5mp2b 10 . . . 4  |-  U. ran  x  e.  _V
7 nlim0 4798 . . . . . 6  |-  -.  Lim  (/)
8 dm0 5074 . . . . . . 7  |-  dom  (/)  =  (/)
9 limeq 4752 . . . . . . 7  |-  ( dom  (/)  =  (/)  ->  ( Lim 
dom  (/)  <->  Lim  (/) ) )
108, 9ax-mp 5 . . . . . 6  |-  ( Lim 
dom  (/)  <->  Lim  (/) )
117, 10mtbir 299 . . . . 5  |-  -.  Lim  dom  (/)
12 dmeq 5061 . . . . . . 7  |-  ( x  =  (/)  ->  dom  x  =  dom  (/) )
13 limeq 4752 . . . . . . 7  |-  ( dom  x  =  dom  (/)  ->  ( Lim  dom  x  <->  Lim  dom  (/) ) )
1412, 13syl 16 . . . . . 6  |-  ( x  =  (/)  ->  ( Lim 
dom  x  <->  Lim  dom  (/) ) )
1514biimpa 484 . . . . 5  |-  ( ( x  =  (/)  /\  Lim  dom  x )  ->  Lim  dom  (/) )
1611, 15mto 176 . . . 4  |-  -.  (
x  =  (/)  /\  Lim  dom  x )
172, 6, 16moeq3 3157 . . 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 1595 . 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 5459 . 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 368    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756   E*wmo 2254   _Vcvv 2993   (/)c0 3658   U.cuni 4112   {copab 4370   Lim wlim 4741   dom cdm 4861   ran crn 4862   Fun wfun 5433   ` cfv 5439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-lim 4745  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-iota 5402  df-fun 5441  df-fv 5447
This theorem is referenced by: (None)
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