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Theorem wfrlem5 29543
Description: Lemma for well-founded recursion. The values of two acceptable functions agree within their domains. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfrlem5.1  |-  R  We  A
wfrlem5.2  |-  R Se  A
wfrlem5.3  |-  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 )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }
Assertion
Ref Expression
wfrlem5  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
Distinct variable groups:    A, f,
g, h, x, y   
f, F, g, h, x, y    R, f, g, h, x, y   
u, g, v, h, x
Allowed substitution hints:    A( v, u)    B( x, y, v, u, f, g, h)    R( v, u)    F( v, u)

Proof of Theorem wfrlem5
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 vex 3112 . . . . . 6  |-  x  e. 
_V
2 vex 3112 . . . . . 6  |-  u  e. 
_V
31, 2breldm 5217 . . . . 5  |-  ( x g u  ->  x  e.  dom  g )
4 vex 3112 . . . . . 6  |-  v  e. 
_V
51, 4breldm 5217 . . . . 5  |-  ( x h v  ->  x  e.  dom  h )
63, 5anim12i 566 . . . 4  |-  ( ( x g u  /\  x h v )  ->  ( x  e. 
dom  g  /\  x  e.  dom  h ) )
7 elin 3683 . . . 4  |-  ( x  e.  ( dom  g  i^i  dom  h )  <->  ( x  e.  dom  g  /\  x  e.  dom  h ) )
86, 7sylibr 212 . . 3  |-  ( ( x g u  /\  x h v )  ->  x  e.  ( dom  g  i^i  dom  h ) )
9 anandir 829 . . . . 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 5290 . . . . . 6  |-  ( x ( g  |`  ( dom  g  i^i  dom  h
) ) u  <->  ( x
g u  /\  x  e.  ( dom  g  i^i 
dom  h ) ) )
114brres 5290 . . . . . 6  |-  ( x ( h  |`  ( dom  g  i^i  dom  h
) ) v  <->  ( x h v  /\  x  e.  ( dom  g  i^i 
dom  h ) ) )
1210, 11anbi12i 697 . . . . 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 252 . . . 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 194 . . 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 668 . 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 wfrlem5.3 . . . . . . . . 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 )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }
1716wfrlem3 29541 . . . . . . . 8  |-  ( g  e.  B  ->  dom  g  C_  A )
18 ssinss1 3722 . . . . . . . 8  |-  ( dom  g  C_  A  ->  ( dom  g  i^i  dom  h )  C_  A
)
19 wfrlem5.1 . . . . . . . . . 10  |-  R  We  A
20 wess 4875 . . . . . . . . . 10  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  ( R  We  A  ->  R  We  ( dom  g  i^i  dom  h
) ) )
2119, 20mpi 17 . . . . . . . . 9  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  R  We  ( dom  g  i^i  dom  h
) )
22 wfrlem5.2 . . . . . . . . . 10  |-  R Se  A
23 sess2 4857 . . . . . . . . . 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 532 . . . . . . . 8  |-  ( ( dom  g  i^i  dom  h )  C_  A  ->  ( R  We  ( dom  g  i^i  dom  h
)  /\  R Se  ( dom  g  i^i  dom  h
) ) )
2617, 18, 253syl 20 . . . . . . 7  |-  ( g  e.  B  ->  ( R  We  ( dom  g  i^i  dom  h )  /\  R Se  ( dom  g  i^i  dom  h )
) )
2726adantr 465 . . . . . 6  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( R  We  ( dom  g  i^i  dom  h
)  /\  R Se  ( dom  g  i^i  dom  h
) ) )
2819, 16wfrlem4 29542 . . . . . 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 )  =  ( F `  (
( g  |`  ( dom  g  i^i  dom  h
) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )
2919, 16wfrlem4 29542 . . . . . . . 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 )  =  ( F `  (
( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) ) ) )
3029ancoms 453 . . . . . . 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 )  =  ( F `  (
( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) ) ) )
31 incom 3687 . . . . . . . . . . 11  |-  ( dom  g  i^i  dom  h
)  =  ( dom  h  i^i  dom  g
)
3231reseq2i 5280 . . . . . . . . . 10  |-  ( h  |`  ( dom  g  i^i 
dom  h ) )  =  ( h  |`  ( dom  h  i^i  dom  g ) )
3332fneq1i 5681 . . . . . . . . 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 5682 . . . . . . . . 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 249 . . . . . . . 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 5873 . . . . . . . . . 10  |-  ( ( h  |`  ( dom  g  i^i  dom  h )
) `  a )  =  ( ( h  |`  ( dom  h  i^i 
dom  g ) ) `
 a )
37 predeq2 29443 . . . . . . . . . . . . 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 5 . . . . . . . . . . . 12  |-  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )  = 
Pred ( R , 
( dom  h  i^i  dom  g ) ,  a )
3932, 38reseq12i 5281 . . . . . . . . . . 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 ) )
4039fveq2i 5875 . . . . . . . . . 10  |-  ( F `
 ( ( h  |`  ( dom  g  i^i 
dom  h ) )  |`  Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) ) )  =  ( F `  (
( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) )
4136, 40eqeq12i 2477 . . . . . . . . 9  |-  ( ( ( h  |`  ( dom  g  i^i  dom  h
) ) `  a
)  =  ( F `
 ( ( h  |`  ( dom  g  i^i 
dom  h ) )  |`  Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) ) )  <->  ( (
h  |`  ( dom  h  i^i  dom  g ) ) `
 a )  =  ( F `  (
( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) ) )
4231, 41raleqbii 2902 . . . . . . . 8  |-  ( A. a  e.  ( dom  g  i^i  dom  h )
( ( h  |`  ( dom  g  i^i  dom  h ) ) `  a )  =  ( F `  ( ( 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
)  =  ( F `
 ( ( h  |`  ( dom  h  i^i 
dom  g ) )  |`  Pred ( R , 
( dom  h  i^i  dom  g ) ,  a ) ) ) )
4335, 42anbi12i 697 . . . . . . 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
)  =  ( F `
 ( ( 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 )  =  ( F `  (
( h  |`  ( dom  h  i^i  dom  g
) )  |`  Pred ( R ,  ( dom  h  i^i  dom  g ) ,  a ) ) ) ) )
4430, 43sylibr 212 . . . . . 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 )  =  ( F `  (
( h  |`  ( dom  g  i^i  dom  h
) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) ) ) )
45 wfr3g 29538 . . . . . 6  |-  ( ( ( R  We  ( 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
)  =  ( F `
 ( ( 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 )  =  ( F `  (
( 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 1228 . . . . 5  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( g  |`  ( dom  g  i^i  dom  h
) )  =  ( h  |`  ( dom  g  i^i  dom  h )
) )
4746breqd 4467 . . . 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 223 . . 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 ) )
4916wfrlem2 29540 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
50 funres 5633 . . . . 5  |-  ( Fun  g  ->  Fun  ( g  |`  ( dom  g  i^i 
dom  h ) ) )
51 dffun2 5604 . . . . . 6  |-  ( 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 464 . . . . 5  |-  ( 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 2sp 1867 . . . . . 6  |-  ( 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 ) )
5453sps 1866 . . . . 5  |-  ( 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 ) )
5549, 50, 52, 544syl 21 . . . 4  |-  ( g  e.  B  ->  (
( x ( g  |`  ( dom  g  i^i 
dom  h ) ) u  /\  x ( g  |`  ( dom  g  i^i  dom  h )
) v )  ->  u  =  v )
)
5655adantr 465 . . 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 ) )
5748, 56sylan2d 482 . 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 ) )
5815, 57syl5 32 1  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1393    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442   A.wral 2807    i^i cin 3470    C_ wss 3471   class class class wbr 4456   Se wse 4845    We wwe 4846   dom cdm 5008    |` cres 5010   Rel wrel 5013   Fun wfun 5588    Fn wfn 5589   ` cfv 5594   Predcpred 29439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-pred 29440
This theorem is referenced by:  wfrlem11  29549
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