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Theorem tz7.44-1 6373
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 5444 . . . 4  |-  ( y  =  (/)  ->  ( F `
 y )  =  ( F `  (/) ) )
2 reseq2 4924 . . . . . 6  |-  ( y  =  (/)  ->  ( F  |`  y )  =  ( F  |`  (/) ) )
3 res0 4933 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
42, 3syl6eq 2304 . . . . 5  |-  ( y  =  (/)  ->  ( F  |`  y )  =  (/) )
54fveq2d 5448 . . . 4  |-  ( y  =  (/)  ->  ( G `
 ( F  |`  y ) )  =  ( G `  (/) ) )
61, 5eqeq12d 2270 . . 3  |-  ( y  =  (/)  ->  ( ( F `  y )  =  ( G `  ( F  |`  y ) )  <->  ( F `  (/) )  =  ( G `
 (/) ) ) )
7 tz7.44.2 . . 3  |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )
86, 7vtoclga 2817 . 2  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  ( G `  (/) ) )
9 0ex 4110 . . 3  |-  (/)  e.  _V
10 iftrue 3531 . . . 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 5522 . . 3  |-  ( (/)  e.  _V  ->  ( G `  (/) )  =  A )
149, 13ax-mp 10 . 2  |-  ( G `
 (/) )  =  A
158, 14syl6eq 2304 1  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1619    e. wcel 1621   _Vcvv 2757   (/)c0 3416   ifcif 3525   U.cuni 3787    e. cmpt 4037   Lim wlim 4351   dom cdm 4647   ran crn 4648    |` cres 4649   ` cfv 4659
This theorem is referenced by:  rdg0  6388
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675
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