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Theorem tfrlem5 6844
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 2980 . . 3  |-  g  e. 
_V
31, 2tfrlem3a 6841 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a ) ) ) )
4 vex 2980 . . 3  |-  h  e. 
_V
51, 4tfrlem3a 6841 . 2  |-  ( h  e.  A  <->  E. w  e.  On  ( h  Fn  w  /\  A. a  e.  w  ( h `  a )  =  ( F `  ( h  |`  a ) ) ) )
6 reeanv 2893 . . 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 simp2ll 1055 . . . . . . . . . 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 ) )  ->  g  Fn  z
)
8 simp3l 1016 . . . . . . . . . 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 )
9 fnbr 5518 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  x g u )  ->  x  e.  z )
107, 8, 9syl2anc 661 . . . . . . . . 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 )
11 simp2rl 1057 . . . . . . . . . 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 ) )  ->  h  Fn  w
)
12 simp3r 1017 . . . . . . . . . 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 )
13 fnbr 5518 . . . . . . . . . 10  |-  ( ( h  Fn  w  /\  x h v )  ->  x  e.  w
)
1411, 12, 13syl2anc 661 . . . . . . . . 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
)
1510, 14elind 3545 . . . . . . . 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 ) )
16 onin 4755 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  w  e.  On )  ->  ( z  i^i  w
)  e.  On )
17163ad2ant1 1009 . . . . . . . . 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 )
18 fnfun 5513 . . . . . . . . . . 11  |-  ( g  Fn  z  ->  Fun  g )
197, 18syl 16 . . . . . . . . . 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 )
20 inss1 3575 . . . . . . . . . . 11  |-  ( z  i^i  w )  C_  z
21 fndm 5515 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  dom  g  =  z )
227, 21syl 16 . . . . . . . . . . 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 )
2320, 22syl5sseqr 3410 . . . . . . . . . 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 )
2419, 23jca 532 . . . . . . . . 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 ) )
25 fnfun 5513 . . . . . . . . . . 11  |-  ( h  Fn  w  ->  Fun  h )
2611, 25syl 16 . . . . . . . . . 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 )
27 inss2 3576 . . . . . . . . . . 11  |-  ( z  i^i  w )  C_  w
28 fndm 5515 . . . . . . . . . . . 12  |-  ( h  Fn  w  ->  dom  h  =  w )
2911, 28syl 16 . . . . . . . . . . 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 )
3027, 29syl5sseqr 3410 . . . . . . . . . 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 )
3126, 30jca 532 . . . . . . . . 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 ) )
32 simp2lr 1056 . . . . . . . . . 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 ) ) )
33 ssralv 3421 . . . . . . . . . 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 ) ) ) )
3420, 32, 33mpsyl 63 . . . . . . . . 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 ) ) )
35 simp2rr 1058 . . . . . . . . . 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 ) ) )
36 ssralv 3421 . . . . . . . . . 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 ) ) ) )
3727, 35, 36mpsyl 63 . . . . . . . . 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 ) ) )
3817, 24, 31, 34, 37tfrlem1 6840 . . . . . . . 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 ) )
39 fveq2 5696 . . . . . . . . . 10  |-  ( a  =  x  ->  (
g `  a )  =  ( g `  x ) )
40 fveq2 5696 . . . . . . . . . 10  |-  ( a  =  x  ->  (
h `  a )  =  ( h `  x ) )
4139, 40eqeq12d 2457 . . . . . . . . 9  |-  ( a  =  x  ->  (
( g `  a
)  =  ( h `
 a )  <->  ( g `  x )  =  ( h `  x ) ) )
4241rspcv 3074 . . . . . . . 8  |-  ( x  e.  ( z  i^i  w )  ->  ( A. a  e.  (
z  i^i  w )
( g `  a
)  =  ( h `
 a )  -> 
( g `  x
)  =  ( h `
 x ) ) )
4315, 38, 42sylc 60 . . . . . . 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 5735 . . . . . . . 8  |-  ( Fun  g  ->  ( x
g u  ->  (
g `  x )  =  u ) )
4519, 8, 44sylc 60 . . . . . . 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 5735 . . . . . . . 8  |-  ( Fun  h  ->  ( x h v  ->  (
h `  x )  =  v ) )
4726, 12, 46sylc 60 . . . . . . 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 2483 . . . . . 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 1186 . . . . 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 2846 . . . 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 2840 . . 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 213 . 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 479 1  |-  ( ( g  e.  A  /\  h  e.  A )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721    i^i cin 3332    C_ wss 3333   class class class wbr 4297   Oncon0 4724   dom cdm 4845    |` cres 4847   Fun wfun 5417    Fn wfn 5418   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431
This theorem is referenced by:  tfrlem7  6847
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