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Theorem tz7.44-1 6415
Description: The value of  F at  (/). Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)
Hypotheses
Ref Expression
tz7.44.1  |-  G  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `
 U. dom  x
) ) ) ) )
tz7.44.2  |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )
tz7.44-1.3  |-  A  e. 
_V
Assertion
Ref Expression
tz7.44-1  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  A )
Distinct variable groups:    x, A    x, y, F    y, G    x, H    y, X
Allowed substitution hints:    A( y)    G( x)    H( y)    X( x)

Proof of Theorem tz7.44-1
StepHypRef Expression
1 fveq2 5486 . . . 4  |-  ( y  =  (/)  ->  ( F `
 y )  =  ( F `  (/) ) )
2 reseq2 4950 . . . . . 6  |-  ( y  =  (/)  ->  ( F  |`  y )  =  ( F  |`  (/) ) )
3 res0 4959 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
42, 3syl6eq 2333 . . . . 5  |-  ( y  =  (/)  ->  ( F  |`  y )  =  (/) )
54fveq2d 5490 . . . 4  |-  ( y  =  (/)  ->  ( G `
 ( F  |`  y ) )  =  ( G `  (/) ) )
61, 5eqeq12d 2299 . . 3  |-  ( y  =  (/)  ->  ( ( F `  y )  =  ( G `  ( F  |`  y ) )  <->  ( F `  (/) )  =  ( G `
 (/) ) ) )
7 tz7.44.2 . . 3  |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )
86, 7vtoclga 2851 . 2  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  ( G `  (/) ) )
9 0ex 4152 . . 3  |-  (/)  e.  _V
10 iftrue 3573 . . . 4  |-  ( x  =  (/)  ->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `  U. dom  x ) ) ) )  =  A )
11 tz7.44.1 . . . 4  |-  G  =  ( x  e.  _V  |->  if ( x  =  (/) ,  A ,  if ( Lim  dom  x ,  U. ran  x ,  ( H `  ( x `
 U. dom  x
) ) ) ) )
12 tz7.44-1.3 . . . 4  |-  A  e. 
_V
1310, 11, 12fvmpt 5564 . . 3  |-  ( (/)  e.  _V  ->  ( G `  (/) )  =  A )
149, 13ax-mp 10 . 2  |-  ( G `
 (/) )  =  A
158, 14syl6eq 2333 1  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   _Vcvv 2790   (/)c0 3457   ifcif 3567   U.cuni 3829    e. cmpt 4079   Lim wlim 4393   dom cdm 4689   ran crn 4690    |` cres 4691   ` cfv 5222
This theorem is referenced by:  rdg0  6430
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fv 5230
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