ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlem5 Unicode version

Theorem tfrlem5 6340
Description: Lemma for transfinite recursion. The values of two acceptable functions are the same within their domains. (Contributed by NM, 9-Apr-1995.) (Revised by Mario Carneiro, 24-May-2019.)
Hypothesis
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfrlem5  |-  ( ( g  e.  A  /\  h  e.  A )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
Distinct variable groups:    f, g, x, y, h, u, v, F    A, g, h
Allowed substitution hints:    A( x, y, v, u, f)

Proof of Theorem tfrlem5
Dummy variables  z  a  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . 3  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
2 vex 2755 . . 3  |-  g  e. 
_V
31, 2tfrlem3a 6336 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a ) ) ) )
4 vex 2755 . . 3  |-  h  e. 
_V
51, 4tfrlem3a 6336 . 2  |-  ( h  e.  A  <->  E. w  e.  On  ( h  Fn  w  /\  A. a  e.  w  ( h `  a )  =  ( F `  ( h  |`  a ) ) ) )
6 reeanv 2660 . . 3  |-  ( E. z  e.  On  E. w  e.  On  (
( g  Fn  z  /\  A. a  e.  z  ( g `  a
)  =  ( F `
 ( g  |`  a ) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a )  =  ( F `  ( h  |`  a ) ) ) )  <->  ( E. z  e.  On  ( g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a ) ) )  /\  E. w  e.  On  ( h  Fn  w  /\  A. a  e.  w  ( h `  a )  =  ( F `  ( h  |`  a ) ) ) ) )
7 fveq2 5534 . . . . . . . . 9  |-  ( a  =  x  ->  (
g `  a )  =  ( g `  x ) )
8 fveq2 5534 . . . . . . . . 9  |-  ( a  =  x  ->  (
h `  a )  =  ( h `  x ) )
97, 8eqeq12d 2204 . . . . . . . 8  |-  ( a  =  x  ->  (
( g `  a
)  =  ( h `
 a )  <->  ( g `  x )  =  ( h `  x ) ) )
10 onin 4404 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  w  e.  On )  ->  ( z  i^i  w
)  e.  On )
11103ad2ant1 1020 . . . . . . . . 9  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  ( z  i^i  w )  e.  On )
12 simp2ll 1066 . . . . . . . . . . 11  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  g  Fn  z
)
13 fnfun 5332 . . . . . . . . . . 11  |-  ( g  Fn  z  ->  Fun  g )
1412, 13syl 14 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  Fun  g )
15 inss1 3370 . . . . . . . . . . 11  |-  ( z  i^i  w )  C_  z
16 fndm 5334 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  dom  g  =  z )
1712, 16syl 14 . . . . . . . . . . 11  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  dom  g  =  z )
1815, 17sseqtrrid 3221 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  ( z  i^i  w )  C_  dom  g )
1914, 18jca 306 . . . . . . . . 9  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  ( Fun  g  /\  ( z  i^i  w
)  C_  dom  g ) )
20 simp2rl 1068 . . . . . . . . . . 11  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  h  Fn  w
)
21 fnfun 5332 . . . . . . . . . . 11  |-  ( h  Fn  w  ->  Fun  h )
2220, 21syl 14 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  Fun  h )
23 inss2 3371 . . . . . . . . . . 11  |-  ( z  i^i  w )  C_  w
24 fndm 5334 . . . . . . . . . . . 12  |-  ( h  Fn  w  ->  dom  h  =  w )
2520, 24syl 14 . . . . . . . . . . 11  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  dom  h  =  w )
2623, 25sseqtrrid 3221 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  ( z  i^i  w )  C_  dom  h )
2722, 26jca 306 . . . . . . . . 9  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  ( Fun  h  /\  ( z  i^i  w
)  C_  dom  h ) )
28 simp2lr 1067 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  A. a  e.  z  ( g `  a
)  =  ( F `
 ( g  |`  a ) ) )
29 ssralv 3234 . . . . . . . . . 10  |-  ( ( z  i^i  w ) 
C_  z  ->  ( A. a  e.  z 
( g `  a
)  =  ( F `
 ( g  |`  a ) )  ->  A. a  e.  (
z  i^i  w )
( g `  a
)  =  ( F `
 ( g  |`  a ) ) ) )
3015, 28, 29mpsyl 65 . . . . . . . . 9  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  A. a  e.  ( z  i^i  w ) ( g `  a
)  =  ( F `
 ( g  |`  a ) ) )
31 simp2rr 1069 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) )
32 ssralv 3234 . . . . . . . . . 10  |-  ( ( z  i^i  w ) 
C_  w  ->  ( A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) )  ->  A. a  e.  (
z  i^i  w )
( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )
3323, 31, 32mpsyl 65 . . . . . . . . 9  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  A. a  e.  ( z  i^i  w ) ( h `  a
)  =  ( F `
 ( h  |`  a ) ) )
3411, 19, 27, 30, 33tfrlem1 6334 . . . . . . . 8  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  A. a  e.  ( z  i^i  w ) ( g `  a
)  =  ( h `
 a ) )
35 simp3l 1027 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  x g u )
36 fnbr 5337 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  x g u )  ->  x  e.  z )
3712, 35, 36syl2anc 411 . . . . . . . . 9  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  x  e.  z )
38 simp3r 1028 . . . . . . . . . 10  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  x h v )
39 fnbr 5337 . . . . . . . . . 10  |-  ( ( h  Fn  w  /\  x h v )  ->  x  e.  w
)
4020, 38, 39syl2anc 411 . . . . . . . . 9  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  x  e.  w
)
41 elin 3333 . . . . . . . . 9  |-  ( x  e.  ( z  i^i  w )  <->  ( x  e.  z  /\  x  e.  w ) )
4237, 40, 41sylanbrc 417 . . . . . . . 8  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  x  e.  ( z  i^i  w ) )
439, 34, 42rspcdva 2861 . . . . . . 7  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  ( g `  x )  =  ( h `  x ) )
44 funbrfv 5575 . . . . . . . 8  |-  ( Fun  g  ->  ( x
g u  ->  (
g `  x )  =  u ) )
4514, 35, 44sylc 62 . . . . . . 7  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  ( g `  x )  =  u )
46 funbrfv 5575 . . . . . . . 8  |-  ( Fun  h  ->  ( x h v  ->  (
h `  x )  =  v ) )
4722, 38, 46sylc 62 . . . . . . 7  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  ( h `  x )  =  v )
4843, 45, 473eqtr3d 2230 . . . . . 6  |-  ( ( ( z  e.  On  /\  w  e.  On )  /\  ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  /\  ( x g u  /\  x h v ) )  ->  u  =  v )
49483exp 1204 . . . . 5  |-  ( ( z  e.  On  /\  w  e.  On )  ->  ( ( ( g  Fn  z  /\  A. a  e.  z  (
g `  a )  =  ( F `  ( g  |`  a
) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a
)  =  ( F `
 ( h  |`  a ) ) ) )  ->  ( (
x g u  /\  x h v )  ->  u  =  v ) ) )
5049rexlimdva 2607 . . . 4  |-  ( z  e.  On  ->  ( E. w  e.  On  ( ( g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a ) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a )  =  ( F `  ( h  |`  a ) ) ) )  ->  ( (
x g u  /\  x h v )  ->  u  =  v ) ) )
5150rexlimiv 2601 . . 3  |-  ( E. z  e.  On  E. w  e.  On  (
( g  Fn  z  /\  A. a  e.  z  ( g `  a
)  =  ( F `
 ( g  |`  a ) ) )  /\  ( h  Fn  w  /\  A. a  e.  w  ( h `  a )  =  ( F `  ( h  |`  a ) ) ) )  ->  ( (
x g u  /\  x h v )  ->  u  =  v ) )
526, 51sylbir 135 . 2  |-  ( ( E. z  e.  On  ( g  Fn  z  /\  A. a  e.  z  ( g `  a
)  =  ( F `
 ( g  |`  a ) ) )  /\  E. w  e.  On  ( h  Fn  w  /\  A. a  e.  w  ( h `  a )  =  ( F `  ( h  |`  a ) ) ) )  ->  ( (
x g u  /\  x h v )  ->  u  =  v ) )
533, 5, 52syl2anb 291 1  |-  ( ( g  e.  A  /\  h  e.  A )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469    i^i cin 3143    C_ wss 3144   class class class wbr 4018   Oncon0 4381   dom cdm 4644    |` cres 4646   Fun wfun 5229    Fn wfn 5230   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by:  tfrlem7  6343  tfrexlem  6360
  Copyright terms: Public domain W3C validator