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Theorem tfrlem5 6290
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 2733 . . 3  |-  g  e. 
_V
31, 2tfrlem3a 6286 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a ) ) ) )
4 vex 2733 . . 3  |-  h  e. 
_V
51, 4tfrlem3a 6286 . 2  |-  ( h  e.  A  <->  E. w  e.  On  ( h  Fn  w  /\  A. a  e.  w  ( h `  a )  =  ( F `  ( h  |`  a ) ) ) )
6 reeanv 2639 . . 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 5494 . . . . . . . . 9  |-  ( a  =  x  ->  (
g `  a )  =  ( g `  x ) )
8 fveq2 5494 . . . . . . . . 9  |-  ( a  =  x  ->  (
h `  a )  =  ( h `  x ) )
97, 8eqeq12d 2185 . . . . . . . 8  |-  ( a  =  x  ->  (
( g `  a
)  =  ( h `
 a )  <->  ( g `  x )  =  ( h `  x ) ) )
10 onin 4369 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  w  e.  On )  ->  ( z  i^i  w
)  e.  On )
11103ad2ant1 1013 . . . . . . . . 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 1059 . . . . . . . . . . 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 5293 . . . . . . . . . . 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 3347 . . . . . . . . . . 11  |-  ( z  i^i  w )  C_  z
16 fndm 5295 . . . . . . . . . . . 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 3198 . . . . . . . . . 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 304 . . . . . . . . 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 1061 . . . . . . . . . . 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 5293 . . . . . . . . . . 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 3348 . . . . . . . . . . 11  |-  ( z  i^i  w )  C_  w
24 fndm 5295 . . . . . . . . . . . 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 3198 . . . . . . . . . 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 304 . . . . . . . . 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 1060 . . . . . . . . . 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 3211 . . . . . . . . . 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 1062 . . . . . . . . . 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 3211 . . . . . . . . . 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 6284 . . . . . . . 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 1020 . . . . . . . . . 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 5298 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  x g u )  ->  x  e.  z )
3712, 35, 36syl2anc 409 . . . . . . . . 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 1021 . . . . . . . . . 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 5298 . . . . . . . . . 10  |-  ( ( h  Fn  w  /\  x h v )  ->  x  e.  w
)
4020, 38, 39syl2anc 409 . . . . . . . . 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 3310 . . . . . . . . 9  |-  ( x  e.  ( z  i^i  w )  <->  ( x  e.  z  /\  x  e.  w ) )
4237, 40, 41sylanbrc 415 . . . . . . . 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 2839 . . . . . . 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 5533 . . . . . . . 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 5533 . . . . . . . 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 2211 . . . . . 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 1197 . . . . 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 2587 . . . 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 2581 . . 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 134 . 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 289 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 103    /\ w3a 973    = wceq 1348    e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449    i^i cin 3120    C_ wss 3121   class class class wbr 3987   Oncon0 4346   dom cdm 4609    |` cres 4611   Fun wfun 5190    Fn wfn 5191   ` cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-res 4621  df-iota 5158  df-fun 5198  df-fn 5199  df-fv 5204
This theorem is referenced by:  tfrlem7  6293  tfrexlem  6310
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