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Theorem tz7.44-1 6655
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 5719 . . . 4  |-  ( y  =  (/)  ->  ( F `
 y )  =  ( F `  (/) ) )
2 reseq2 5132 . . . . . 6  |-  ( y  =  (/)  ->  ( F  |`  y )  =  ( F  |`  (/) ) )
3 res0 5141 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
42, 3syl6eq 2483 . . . . 5  |-  ( y  =  (/)  ->  ( F  |`  y )  =  (/) )
54fveq2d 5723 . . . 4  |-  ( y  =  (/)  ->  ( G `
 ( F  |`  y ) )  =  ( G `  (/) ) )
61, 5eqeq12d 2449 . . 3  |-  ( y  =  (/)  ->  ( ( F `  y )  =  ( G `  ( F  |`  y ) )  <->  ( F `  (/) )  =  ( G `
 (/) ) ) )
7 tz7.44.2 . . 3  |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )
86, 7vtoclga 3009 . 2  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  ( G `  (/) ) )
9 0ex 4331 . . 3  |-  (/)  e.  _V
10 iftrue 3737 . . . 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 5797 . . 3  |-  ( (/)  e.  _V  ->  ( G `  (/) )  =  A )
149, 13ax-mp 8 . 2  |-  ( G `
 (/) )  =  A
158, 14syl6eq 2483 1  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   ifcif 3731   U.cuni 4007    e. cmpt 4258   Lim wlim 4574   dom cdm 4869   ran crn 4870    |` cres 4871   ` cfv 5445
This theorem is referenced by:  rdg0  6670
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-res 4881  df-iota 5409  df-fun 5447  df-fv 5453
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