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Theorem tz7.44-1 6419
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 5525 . . . 4  |-  ( y  =  (/)  ->  ( F `
 y )  =  ( F `  (/) ) )
2 reseq2 4950 . . . . . 6  |-  ( y  =  (/)  ->  ( F  |`  y )  =  ( F  |`  (/) ) )
3 res0 4959 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
42, 3syl6eq 2331 . . . . 5  |-  ( y  =  (/)  ->  ( F  |`  y )  =  (/) )
54fveq2d 5529 . . . 4  |-  ( y  =  (/)  ->  ( G `
 ( F  |`  y ) )  =  ( G `  (/) ) )
61, 5eqeq12d 2297 . . 3  |-  ( y  =  (/)  ->  ( ( F `  y )  =  ( G `  ( F  |`  y ) )  <->  ( F `  (/) )  =  ( G `
 (/) ) ) )
7 tz7.44.2 . . 3  |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )
86, 7vtoclga 2849 . 2  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  ( G `  (/) ) )
9 0ex 4150 . . 3  |-  (/)  e.  _V
10 iftrue 3571 . . . 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 5602 . . 3  |-  ( (/)  e.  _V  ->  ( G `  (/) )  =  A )
149, 13ax-mp 8 . 2  |-  ( G `
 (/) )  =  A
158, 14syl6eq 2331 1  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   ifcif 3565   U.cuni 3827    e. cmpt 4077   Lim wlim 4393   dom cdm 4689   ran crn 4690    |` cres 4691   ` cfv 5255
This theorem is referenced by:  rdg0  6434
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-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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263
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