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Theorem tz7.44lem1 7061
Description:  G is a function. Lemma for tz7.44-1 7062, tz7.44-2 7063, and tz7.44-3 7064. (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 5612 . . 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 5867 . . . 4  |-  ( H `
 ( x `  U. dom  x ) )  e.  _V
3 vex 3109 . . . . 5  |-  x  e. 
_V
4 rnexg 6706 . . . . 5  |-  ( x  e.  _V  ->  ran  x  e.  _V )
5 uniexg 6572 . . . . 5  |-  ( ran  x  e.  _V  ->  U.
ran  x  e.  _V )
63, 4, 5mp2b 10 . . . 4  |-  U. ran  x  e.  _V
7 nlim0 4929 . . . . . 6  |-  -.  Lim  (/)
8 dm0 5207 . . . . . . 7  |-  dom  (/)  =  (/)
9 limeq 4883 . . . . . . 7  |-  ( dom  (/)  =  (/)  ->  ( Lim 
dom  (/)  <->  Lim  (/) ) )
108, 9ax-mp 5 . . . . . 6  |-  ( Lim 
dom  (/)  <->  Lim  (/) )
117, 10mtbir 299 . . . . 5  |-  -.  Lim  dom  (/)
12 dmeq 5194 . . . . . . 7  |-  ( x  =  (/)  ->  dom  x  =  dom  (/) )
13 limeq 4883 . . . . . . 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 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 1600 . 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 5599 . 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 967    = wceq 1374    e. wcel 1762   E*wmo 2269   _Vcvv 3106   (/)c0 3778   U.cuni 4238   {copab 4497   Lim wlim 4872   dom cdm 4992   ran crn 4993   Fun wfun 5573   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-lim 4876  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-iota 5542  df-fun 5581  df-fv 5587
This theorem is referenced by: (None)
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