Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frrlem5 Unicode version

Theorem frrlem5 24356
Description: Lemma for founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
frrlem5.1  |-  R  Fr  A
frrlem5.2  |-  R Se  A
frrlem5.3  |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }
Assertion
Ref Expression
frrlem5  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
Distinct variable groups:    A, f,
g, h, x, y   
f, G, h, x, y, g    u, g, v, x    y, g   
u, h, v    R, f, g, h, x, y    B, g, h, u, v, x
Allowed substitution hints:    A( v, u)    B( y, f)    R( v, u)    G( v, u)

Proof of Theorem frrlem5
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 vex 2804 . . . . . 6  |-  x  e. 
_V
2 vex 2804 . . . . . 6  |-  u  e. 
_V
31, 2breldm 4899 . . . . 5  |-  ( x g u  ->  x  e.  dom  g )
4 vex 2804 . . . . . 6  |-  v  e. 
_V
51, 4breldm 4899 . . . . 5  |-  ( x h v  ->  x  e.  dom  h )
63, 5anim12i 549 . . . 4  |-  ( ( x g u  /\  x h v )  ->  ( x  e. 
dom  g  /\  x  e.  dom  h ) )
7 elin 3371 . . . 4  |-  ( x  e.  ( dom  g  i^i  dom  h )  <->  ( x  e.  dom  g  /\  x  e.  dom  h ) )
86, 7sylibr 203 . . 3  |-  ( ( x g u  /\  x h v )  ->  x  e.  ( dom  g  i^i  dom  h ) )
9 anandir 802 . . . . 5  |-  ( ( ( x g u  /\  x h v )  /\  x  e.  ( dom  g  i^i 
dom  h ) )  <-> 
( ( x g u  /\  x  e.  ( dom  g  i^i 
dom  h ) )  /\  ( x h v  /\  x  e.  ( dom  g  i^i 
dom  h ) ) ) )
102brres 4977 . . . . . 6  |-  ( x ( g  |`  ( dom  g  i^i  dom  h
) ) u  <->  ( x
g u  /\  x  e.  ( dom  g  i^i 
dom  h ) ) )
114brres 4977 . . . . . 6  |-  ( x ( h  |`  ( dom  g  i^i  dom  h
) ) v  <->  ( x h v  /\  x  e.  ( dom  g  i^i 
dom  h ) ) )
1210, 11anbi12i 678 . . . . 5  |-  ( ( x ( g  |`  ( dom  g  i^i  dom  h ) ) u  /\  x ( h  |`  ( dom  g  i^i 
dom  h ) ) v )  <->  ( (
x g u  /\  x  e.  ( dom  g  i^i  dom  h )
)  /\  ( x h v  /\  x  e.  ( dom  g  i^i 
dom  h ) ) ) )
139, 12bitr4i 243 . . . 4  |-  ( ( ( x g u  /\  x h v )  /\  x  e.  ( dom  g  i^i 
dom  h ) )  <-> 
( x ( g  |`  ( dom  g  i^i 
dom  h ) ) u  /\  x ( h  |`  ( dom  g  i^i  dom  h )
) v ) )
1413biimpi 186 . . 3  |-  ( ( ( x g u  /\  x h v )  /\  x  e.  ( dom  g  i^i 
dom  h ) )  ->  ( x ( g  |`  ( dom  g  i^i  dom  h )
) u  /\  x
( h  |`  ( dom  g  i^i  dom  h
) ) v ) )
158, 14mpdan 649 . 2  |-  ( ( x g u  /\  x h v )  ->  ( x ( g  |`  ( dom  g  i^i  dom  h )
) u  /\  x
( h  |`  ( dom  g  i^i  dom  h
) ) v ) )
16 frrlem5.3 . . . . . . . . 9  |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }
1716frrlem3 24354 . . . . . . . 8  |-  ( g  e.  B  ->  dom  g  C_  A )
18 ssinss1 3410 . . . . . . . 8  |-  ( dom  g  C_  A  ->  ( dom  g  i^i  dom  h )  C_  A
)
19 frrlem5.1 . . . . . . . . . 10  |-  R  Fr  A
20 frss 4376 . . . . . . . . . 10  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  ( R  Fr  A  ->  R  Fr  ( dom  g  i^i  dom  h
) ) )
2119, 20mpi 16 . . . . . . . . 9  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  R  Fr  ( dom  g  i^i  dom  h
) )
22 frrlem5.2 . . . . . . . . . 10  |-  R Se  A
23 sess2 4378 . . . . . . . . . 10  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  ( R Se  A  ->  R Se  ( dom  g  i^i 
dom  h ) ) )
2422, 23mpi 16 . . . . . . . . 9  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  R Se  ( dom  g  i^i  dom  h ) )
2521, 24jca 518 . . . . . . . 8  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  ( R  Fr  ( dom  g  i^i  dom  h
)  /\  R Se  ( dom  g  i^i  dom  h
) ) )
2617, 18, 253syl 18 . . . . . . 7  |-  ( g  e.  B  ->  ( R  Fr  ( dom  g  i^i  dom  h )  /\  R Se  ( dom  g  i^i  dom  h )
) )
2726adantr 451 . . . . . 6  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( R  Fr  ( dom  g  i^i  dom  h
)  /\  R Se  ( dom  g  i^i  dom  h
) ) )
2816frrlem4 24355 . . . . . 6  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( g  |`  ( dom  g  i^i  dom  h ) )  Fn  ( dom  g  i^i 
dom  h )  /\  A. a  e.  ( dom  g  i^i  dom  h
) ( ( g  |`  ( dom  g  i^i 
dom  h ) ) `
 a )  =  ( a G ( ( g  |`  ( dom  g  i^i  dom  h
) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )
2916frrlem4 24355 . . . . . . . 8  |-  ( ( h  e.  B  /\  g  e.  B )  ->  ( ( h  |`  ( dom  h  i^i  dom  g ) )  Fn  ( dom  h  i^i 
dom  g )  /\  A. a  e.  ( dom  h  i^i  dom  g
) ( ( h  |`  ( dom  h  i^i 
dom  g ) ) `
 a )  =  ( a G ( ( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) ) ) )
3029ancoms 439 . . . . . . 7  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( h  |`  ( dom  h  i^i  dom  g ) )  Fn  ( dom  h  i^i 
dom  g )  /\  A. a  e.  ( dom  h  i^i  dom  g
) ( ( h  |`  ( dom  h  i^i 
dom  g ) ) `
 a )  =  ( a G ( ( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) ) ) )
31 incom 3374 . . . . . . . . . . 11  |-  ( dom  g  i^i  dom  h
)  =  ( dom  h  i^i  dom  g
)
3231reseq2i 4968 . . . . . . . . . 10  |-  ( h  |`  ( dom  g  i^i 
dom  h ) )  =  ( h  |`  ( dom  h  i^i  dom  g ) )
3332fneq1i 5354 . . . . . . . . 9  |-  ( ( h  |`  ( dom  g  i^i  dom  h )
)  Fn  ( dom  g  i^i  dom  h
)  <->  ( h  |`  ( dom  h  i^i  dom  g ) )  Fn  ( dom  g  i^i 
dom  h ) )
3431fneq2i 5355 . . . . . . . . 9  |-  ( ( h  |`  ( dom  h  i^i  dom  g )
)  Fn  ( dom  g  i^i  dom  h
)  <->  ( h  |`  ( dom  h  i^i  dom  g ) )  Fn  ( dom  h  i^i 
dom  g ) )
3533, 34bitri 240 . . . . . . . 8  |-  ( ( h  |`  ( dom  g  i^i  dom  h )
)  Fn  ( dom  g  i^i  dom  h
)  <->  ( h  |`  ( dom  h  i^i  dom  g ) )  Fn  ( dom  h  i^i 
dom  g ) )
3631eleq2i 2360 . . . . . . . . . 10  |-  ( a  e.  ( dom  g  i^i  dom  h )  <->  a  e.  ( dom  h  i^i  dom  g ) )
3732fveq1i 5542 . . . . . . . . . . 11  |-  ( ( h  |`  ( dom  g  i^i  dom  h )
) `  a )  =  ( ( h  |`  ( dom  h  i^i 
dom  g ) ) `
 a )
38 predeq2 24241 . . . . . . . . . . . . . 14  |-  ( ( dom  g  i^i  dom  h )  =  ( dom  h  i^i  dom  g )  ->  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )  = 
Pred ( R , 
( dom  h  i^i  dom  g ) ,  a ) )
3931, 38ax-mp 8 . . . . . . . . . . . . 13  |-  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )  = 
Pred ( R , 
( dom  h  i^i  dom  g ) ,  a )
4032, 39reseq12i 4969 . . . . . . . . . . . 12  |-  ( ( h  |`  ( dom  g  i^i  dom  h )
)  |`  Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) )  =  ( ( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) )
4140oveq2i 5885 . . . . . . . . . . 11  |-  ( a G ( ( h  |`  ( dom  g  i^i 
dom  h ) )  |`  Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) ) )  =  ( a G ( ( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) )
4237, 41eqeq12i 2309 . . . . . . . . . 10  |-  ( ( ( h  |`  ( dom  g  i^i  dom  h
) ) `  a
)  =  ( a G ( ( h  |`  ( dom  g  i^i 
dom  h ) )  |`  Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) ) )  <->  ( (
h  |`  ( dom  h  i^i  dom  g ) ) `
 a )  =  ( a G ( ( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) ) )
4336, 42imbi12i 316 . . . . . . . . 9  |-  ( ( a  e.  ( dom  g  i^i  dom  h
)  ->  ( (
h  |`  ( dom  g  i^i  dom  h ) ) `
 a )  =  ( a G ( ( h  |`  ( dom  g  i^i  dom  h
) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) )  <->  ( a  e.  ( dom  h  i^i 
dom  g )  -> 
( ( h  |`  ( dom  h  i^i  dom  g ) ) `  a )  =  ( a G ( ( h  |`  ( dom  h  i^i  dom  g )
)  |`  Pred ( R , 
( dom  h  i^i  dom  g ) ,  a ) ) ) ) )
4443ralbii2 2584 . . . . . . . 8  |-  ( A. a  e.  ( dom  g  i^i  dom  h )
( ( h  |`  ( dom  g  i^i  dom  h ) ) `  a )  =  ( a G ( ( h  |`  ( dom  g  i^i  dom  h )
)  |`  Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) ) )  <->  A. a  e.  ( dom  h  i^i 
dom  g ) ( ( h  |`  ( dom  h  i^i  dom  g
) ) `  a
)  =  ( a G ( ( h  |`  ( dom  h  i^i 
dom  g ) )  |`  Pred ( R , 
( dom  h  i^i  dom  g ) ,  a ) ) ) )
4535, 44anbi12i 678 . . . . . . 7  |-  ( ( ( h  |`  ( dom  g  i^i  dom  h
) )  Fn  ( dom  g  i^i  dom  h
)  /\  A. a  e.  ( dom  g  i^i 
dom  h ) ( ( h  |`  ( dom  g  i^i  dom  h
) ) `  a
)  =  ( a G ( ( h  |`  ( dom  g  i^i 
dom  h ) )  |`  Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) ) ) )  <-> 
( ( h  |`  ( dom  h  i^i  dom  g ) )  Fn  ( dom  h  i^i 
dom  g )  /\  A. a  e.  ( dom  h  i^i  dom  g
) ( ( h  |`  ( dom  h  i^i 
dom  g ) ) `
 a )  =  ( a G ( ( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) ) ) )
4630, 45sylibr 203 . . . . . 6  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( h  |`  ( dom  g  i^i  dom  h ) )  Fn  ( dom  g  i^i 
dom  h )  /\  A. a  e.  ( dom  g  i^i  dom  h
) ( ( h  |`  ( dom  g  i^i 
dom  h ) ) `
 a )  =  ( a G ( ( h  |`  ( dom  g  i^i  dom  h
) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )
47 frr3g 24351 . . . . . 6  |-  ( ( ( R  Fr  ( dom  g  i^i  dom  h
)  /\  R Se  ( dom  g  i^i  dom  h
) )  /\  (
( g  |`  ( dom  g  i^i  dom  h
) )  Fn  ( dom  g  i^i  dom  h
)  /\  A. a  e.  ( dom  g  i^i 
dom  h ) ( ( g  |`  ( dom  g  i^i  dom  h
) ) `  a
)  =  ( a G ( ( g  |`  ( dom  g  i^i 
dom  h ) )  |`  Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) ) ) )  /\  ( ( h  |`  ( dom  g  i^i 
dom  h ) )  Fn  ( dom  g  i^i  dom  h )  /\  A. a  e.  ( dom  g  i^i  dom  h
) ( ( h  |`  ( dom  g  i^i 
dom  h ) ) `
 a )  =  ( a G ( ( h  |`  ( dom  g  i^i  dom  h
) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )  -> 
( g  |`  ( dom  g  i^i  dom  h
) )  =  ( h  |`  ( dom  g  i^i  dom  h )
) )
4827, 28, 46, 47syl3anc 1182 . . . . 5  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( g  |`  ( dom  g  i^i  dom  h
) )  =  ( h  |`  ( dom  g  i^i  dom  h )
) )
4948breqd 4050 . . . 4  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( x ( g  |`  ( dom  g  i^i 
dom  h ) ) v  <->  x ( h  |`  ( dom  g  i^i 
dom  h ) ) v ) )
5049biimprd 214 . . 3  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( x ( h  |`  ( dom  g  i^i 
dom  h ) ) v  ->  x (
g  |`  ( dom  g  i^i  dom  h ) ) v ) )
5116frrlem2 24353 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
52 funres 5309 . . . . 5  |-  ( Fun  g  ->  Fun  ( g  |`  ( dom  g  i^i 
dom  h ) ) )
53 dffun2 5281 . . . . . . 7  |-  ( Fun  ( g  |`  ( dom  g  i^i  dom  h
) )  <->  ( Rel  ( g  |`  ( dom  g  i^i  dom  h
) )  /\  A. x A. u A. v
( ( x ( g  |`  ( dom  g  i^i  dom  h )
) u  /\  x
( g  |`  ( dom  g  i^i  dom  h
) ) v )  ->  u  =  v ) ) )
5453simprbi 450 . . . . . 6  |-  ( Fun  ( g  |`  ( dom  g  i^i  dom  h
) )  ->  A. x A. u A. v ( ( x ( g  |`  ( dom  g  i^i 
dom  h ) ) u  /\  x ( g  |`  ( dom  g  i^i  dom  h )
) v )  ->  u  =  v )
)
55 sp 1728 . . . . . . . 8  |-  ( A. v ( ( x ( g  |`  ( dom  g  i^i  dom  h
) ) u  /\  x ( g  |`  ( dom  g  i^i  dom  h ) ) v )  ->  u  =  v )  ->  (
( x ( g  |`  ( dom  g  i^i 
dom  h ) ) u  /\  x ( g  |`  ( dom  g  i^i  dom  h )
) v )  ->  u  =  v )
)
5655sps 1751 . . . . . . 7  |-  ( A. u A. v ( ( x ( g  |`  ( dom  g  i^i  dom  h ) ) u  /\  x ( g  |`  ( dom  g  i^i 
dom  h ) ) v )  ->  u  =  v )  -> 
( ( x ( g  |`  ( dom  g  i^i  dom  h )
) u  /\  x
( g  |`  ( dom  g  i^i  dom  h
) ) v )  ->  u  =  v ) )
5756sps 1751 . . . . . 6  |-  ( A. x A. u A. v
( ( x ( g  |`  ( dom  g  i^i  dom  h )
) u  /\  x
( g  |`  ( dom  g  i^i  dom  h
) ) v )  ->  u  =  v )  ->  ( (
x ( g  |`  ( dom  g  i^i  dom  h ) ) u  /\  x ( g  |`  ( dom  g  i^i 
dom  h ) ) v )  ->  u  =  v ) )
5854, 57syl 15 . . . . 5  |-  ( Fun  ( g  |`  ( dom  g  i^i  dom  h
) )  ->  (
( x ( g  |`  ( dom  g  i^i 
dom  h ) ) u  /\  x ( g  |`  ( dom  g  i^i  dom  h )
) v )  ->  u  =  v )
)
5951, 52, 583syl 18 . . . 4  |-  ( g  e.  B  ->  (
( x ( g  |`  ( dom  g  i^i 
dom  h ) ) u  /\  x ( g  |`  ( dom  g  i^i  dom  h )
) v )  ->  u  =  v )
)
6059adantr 451 . . 3  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( x ( g  |`  ( dom  g  i^i  dom  h )
) u  /\  x
( g  |`  ( dom  g  i^i  dom  h
) ) v )  ->  u  =  v ) )
6150, 60sylan2d 468 . 2  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( x ( g  |`  ( dom  g  i^i  dom  h )
) u  /\  x
( h  |`  ( dom  g  i^i  dom  h
) ) v )  ->  u  =  v ) )
6215, 61syl5 28 1  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556    i^i cin 3164    C_ wss 3165   class class class wbr 4039    Fr wfr 4365   Se wse 4366   dom cdm 4705    |` cres 4707   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   Predcpred 24238
This theorem is referenced by:  frrlem5c  24358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439  df-pred 24239  df-trpred 24292
  Copyright terms: Public domain W3C validator