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Theorem tfrlem5 7046
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 3116 . . 3  |-  g  e. 
_V
31, 2tfrlem3a 7043 . 2  |-  ( g  e.  A  <->  E. z  e.  On  ( g  Fn  z  /\  A. a  e.  z  ( g `  a )  =  ( F `  ( g  |`  a ) ) ) )
4 vex 3116 . . 3  |-  h  e. 
_V
51, 4tfrlem3a 7043 . 2  |-  ( h  e.  A  <->  E. w  e.  On  ( h  Fn  w  /\  A. a  e.  w  ( h `  a )  =  ( F `  ( h  |`  a ) ) ) )
6 reeanv 3029 . . 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 1063 . . . . . . . . . 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 1024 . . . . . . . . . 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 5681 . . . . . . . . . 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 1065 . . . . . . . . . 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 1025 . . . . . . . . . 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 5681 . . . . . . . . . 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 3688 . . . . . . . 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 4909 . . . . . . . . . 10  |-  ( ( z  e.  On  /\  w  e.  On )  ->  ( z  i^i  w
)  e.  On )
17163ad2ant1 1017 . . . . . . . . 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 5676 . . . . . . . . . . 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 3718 . . . . . . . . . . 11  |-  ( z  i^i  w )  C_  z
21 fndm 5678 . . . . . . . . . . . 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 3553 . . . . . . . . . 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 5676 . . . . . . . . . . 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 3719 . . . . . . . . . . 11  |-  ( z  i^i  w )  C_  w
28 fndm 5678 . . . . . . . . . . . 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 3553 . . . . . . . . . 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 1064 . . . . . . . . . 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 3564 . . . . . . . . . 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 1066 . . . . . . . . . 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 3564 . . . . . . . . . 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 7042 . . . . . . . 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 5864 . . . . . . . . . 10  |-  ( a  =  x  ->  (
g `  a )  =  ( g `  x ) )
40 fveq2 5864 . . . . . . . . . 10  |-  ( a  =  x  ->  (
h `  a )  =  ( h `  x ) )
4139, 40eqeq12d 2489 . . . . . . . . 9  |-  ( a  =  x  ->  (
( g `  a
)  =  ( h `
 a )  <->  ( g `  x )  =  ( h `  x ) ) )
4241rspcv 3210 . . . . . . . 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 5904 . . . . . . . 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 5904 . . . . . . . 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 2516 . . . . . 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 1195 . . . . 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 2955 . . . 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 2949 . . 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 973    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815    i^i cin 3475    C_ wss 3476   class class class wbr 4447   Oncon0 4878   dom cdm 4999    |` cres 5001   Fun wfun 5580    Fn wfn 5581   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  tfrlem7  7049
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