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Theorem frrlem5 25302
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 2895 . . . . . 6  |-  x  e. 
_V
2 vex 2895 . . . . . 6  |-  u  e. 
_V
31, 2breldm 5007 . . . . 5  |-  ( x g u  ->  x  e.  dom  g )
4 vex 2895 . . . . . 6  |-  v  e. 
_V
51, 4breldm 5007 . . . . 5  |-  ( x h v  ->  x  e.  dom  h )
63, 5anim12i 550 . . . 4  |-  ( ( x g u  /\  x h v )  ->  ( x  e. 
dom  g  /\  x  e.  dom  h ) )
7 elin 3466 . . . 4  |-  ( x  e.  ( dom  g  i^i  dom  h )  <->  ( x  e.  dom  g  /\  x  e.  dom  h ) )
86, 7sylibr 204 . . 3  |-  ( ( x g u  /\  x h v )  ->  x  e.  ( dom  g  i^i  dom  h ) )
9 anandir 803 . . . . 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 5085 . . . . . 6  |-  ( x ( g  |`  ( dom  g  i^i  dom  h
) ) u  <->  ( x
g u  /\  x  e.  ( dom  g  i^i 
dom  h ) ) )
114brres 5085 . . . . . 6  |-  ( x ( h  |`  ( dom  g  i^i  dom  h
) ) v  <->  ( x h v  /\  x  e.  ( dom  g  i^i 
dom  h ) ) )
1210, 11anbi12i 679 . . . . 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 244 . . . 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 187 . . 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 650 . 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 25300 . . . . . . . 8  |-  ( g  e.  B  ->  dom  g  C_  A )
18 ssinss1 3505 . . . . . . . 8  |-  ( dom  g  C_  A  ->  ( dom  g  i^i  dom  h )  C_  A
)
19 frrlem5.1 . . . . . . . . . 10  |-  R  Fr  A
20 frss 4483 . . . . . . . . . 10  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  ( R  Fr  A  ->  R  Fr  ( dom  g  i^i  dom  h
) ) )
2119, 20mpi 17 . . . . . . . . 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 4485 . . . . . . . . . 10  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  ( R Se  A  ->  R Se  ( dom  g  i^i 
dom  h ) ) )
2422, 23mpi 17 . . . . . . . . 9  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  R Se  ( dom  g  i^i  dom  h ) )
2521, 24jca 519 . . . . . . . 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 19 . . . . . . 7  |-  ( g  e.  B  ->  ( R  Fr  ( dom  g  i^i  dom  h )  /\  R Se  ( dom  g  i^i  dom  h )
) )
2726adantr 452 . . . . . 6  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( R  Fr  ( dom  g  i^i  dom  h
)  /\  R Se  ( dom  g  i^i  dom  h
) ) )
2816frrlem4 25301 . . . . . 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 25301 . . . . . . . 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 440 . . . . . . 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 3469 . . . . . . . . . . 11  |-  ( dom  g  i^i  dom  h
)  =  ( dom  h  i^i  dom  g
)
3231reseq2i 5076 . . . . . . . . . 10  |-  ( h  |`  ( dom  g  i^i 
dom  h ) )  =  ( h  |`  ( dom  h  i^i  dom  g ) )
3332fneq1i 5472 . . . . . . . . 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 5473 . . . . . . . . 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 241 . . . . . . . 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 ) )
3632fveq1i 5662 . . . . . . . . . 10  |-  ( ( h  |`  ( dom  g  i^i  dom  h )
) `  a )  =  ( ( h  |`  ( dom  h  i^i 
dom  g ) ) `
 a )
37 predeq2 25187 . . . . . . . . . . . . 13  |-  ( ( 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 ) )
3831, 37ax-mp 8 . . . . . . . . . . . 12  |-  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )  = 
Pred ( R , 
( dom  h  i^i  dom  g ) ,  a )
3932, 38reseq12i 5077 . . . . . . . . . . 11  |-  ( ( 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 ) )
4039oveq2i 6024 . . . . . . . . . 10  |-  ( 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 ) ) )
4136, 40eqeq12i 2393 . . . . . . . . 9  |-  ( ( ( 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 ) ) ) )
4231, 41raleqbii 2672 . . . . . . . 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 ) ) ) )
4335, 42anbi12i 679 . . . . . . 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 ) ) ) ) )
4430, 43sylibr 204 . . . . . 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 ) ) ) ) )
45 frr3g 25297 . . . . . 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 )
) )
4627, 28, 44, 45syl3anc 1184 . . . . 5  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( g  |`  ( dom  g  i^i  dom  h
) )  =  ( h  |`  ( dom  g  i^i  dom  h )
) )
4746breqd 4157 . . . 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 ) )
4847biimprd 215 . . 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 ) )
4916frrlem2 25299 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
50 funres 5425 . . . . 5  |-  ( Fun  g  ->  Fun  ( g  |`  ( dom  g  i^i 
dom  h ) ) )
51 dffun2 5397 . . . . . . 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 ) ) )
5251simprbi 451 . . . . . 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 )
)
53 sp 1755 . . . . . . . 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 )
)
5453sps 1762 . . . . . . 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 ) )
5554sps 1762 . . . . . 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 ) )
5652, 55syl 16 . . . . 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 )
)
5749, 50, 563syl 19 . . . 4  |-  ( g  e.  B  ->  (
( x ( g  |`  ( dom  g  i^i 
dom  h ) ) u  /\  x ( g  |`  ( dom  g  i^i  dom  h )
) v )  ->  u  =  v )
)
5857adantr 452 . . 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 ) )
5948, 58sylan2d 469 . 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 ) )
6015, 59syl5 30 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 359    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2366   A.wral 2642    i^i cin 3255    C_ wss 3256   class class class wbr 4146    Fr wfr 4472   Se wse 4473   dom cdm 4811    |` cres 4813   Rel wrel 4816   Fun wfun 5381    Fn wfn 5382   ` cfv 5387  (class class class)co 6013   Predcpred 25184
This theorem is referenced by:  frrlem5c  25304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-recs 6562  df-rdg 6597  df-pred 25185  df-trpred 25238
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