MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.44-1 Structured version   Unicode version

Theorem tz7.44-1 6964
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 5791 . . . 4  |-  ( y  =  (/)  ->  ( F `
 y )  =  ( F `  (/) ) )
2 reseq2 5205 . . . . . 6  |-  ( y  =  (/)  ->  ( F  |`  y )  =  ( F  |`  (/) ) )
3 res0 5215 . . . . . 6  |-  ( F  |`  (/) )  =  (/)
42, 3syl6eq 2508 . . . . 5  |-  ( y  =  (/)  ->  ( F  |`  y )  =  (/) )
54fveq2d 5795 . . . 4  |-  ( y  =  (/)  ->  ( G `
 ( F  |`  y ) )  =  ( G `  (/) ) )
61, 5eqeq12d 2473 . . 3  |-  ( y  =  (/)  ->  ( ( F `  y )  =  ( G `  ( F  |`  y ) )  <->  ( F `  (/) )  =  ( G `
 (/) ) ) )
7 tz7.44.2 . . 3  |-  ( y  e.  X  ->  ( F `  y )  =  ( G `  ( F  |`  y ) ) )
86, 7vtoclga 3134 . 2  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  ( G `  (/) ) )
9 0ex 4522 . . 3  |-  (/)  e.  _V
10 iftrue 3897 . . . 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 5875 . . 3  |-  ( (/)  e.  _V  ->  ( G `  (/) )  =  A )
149, 13ax-mp 5 . 2  |-  ( G `
 (/) )  =  A
158, 14syl6eq 2508 1  |-  ( (/)  e.  X  ->  ( F `
 (/) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3070   (/)c0 3737   ifcif 3891   U.cuni 4191    |-> cmpt 4450   Lim wlim 4820   dom cdm 4940   ran crn 4941    |` cres 4942   ` cfv 5518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-res 4952  df-iota 5481  df-fun 5520  df-fv 5526
This theorem is referenced by:  rdg0  6979
  Copyright terms: Public domain W3C validator