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Theorem wfrlem5 28910
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 3109 . . . . . 6  |-  x  e. 
_V
2 vex 3109 . . . . . 6  |-  u  e. 
_V
31, 2breldm 5198 . . . . 5  |-  ( x g u  ->  x  e.  dom  g )
4 vex 3109 . . . . . 6  |-  v  e. 
_V
51, 4breldm 5198 . . . . 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 3680 . . . 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 826 . . . . 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 5271 . . . . . 6  |-  ( x ( g  |`  ( dom  g  i^i  dom  h
) ) u  <->  ( x
g u  /\  x  e.  ( dom  g  i^i 
dom  h ) ) )
114brres 5271 . . . . . 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 28908 . . . . . . . 8  |-  ( g  e.  B  ->  dom  g  C_  A )
18 ssinss1 3719 . . . . . . . 8  |-  ( dom  g  C_  A  ->  ( dom  g  i^i  dom  h )  C_  A
)
19 wfrlem5.1 . . . . . . . . . 10  |-  R  We  A
20 wess 4859 . . . . . . . . . 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 4841 . . . . . . . . . 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 28909 . . . . . 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 28909 . . . . . . . 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 3684 . . . . . . . . . . 11  |-  ( dom  g  i^i  dom  h
)  =  ( dom  h  i^i  dom  g
)
3231reseq2i 5261 . . . . . . . . . 10  |-  ( h  |`  ( dom  g  i^i 
dom  h ) )  =  ( h  |`  ( dom  h  i^i  dom  g ) )
3332fneq1i 5666 . . . . . . . . 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 5667 . . . . . . . . 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 5858 . . . . . . . . . 10  |-  ( ( h  |`  ( dom  g  i^i  dom  h )
) `  a )  =  ( ( h  |`  ( dom  h  i^i 
dom  g ) ) `
 a )
37 predeq2 28810 . . . . . . . . . . . . 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 5262 . . . . . . . . . . 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 5860 . . . . . . . . . 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 2480 . . . . . . . . 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 28905 . . . . . 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 1223 . . . . 5  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( g  |`  ( dom  g  i^i  dom  h
) )  =  ( h  |`  ( dom  g  i^i  dom  h )
) )
4746breqd 4451 . . . 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 28907 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
50 funres 5618 . . . . 5  |-  ( Fun  g  ->  Fun  ( g  |`  ( dom  g  i^i 
dom  h ) ) )
51 dffun2 5589 . . . . . 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 1810 . . . . . 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 1809 . . . . 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 968   A.wal 1372    = wceq 1374   E.wex 1591    e. wcel 1762   {cab 2445   A.wral 2807    i^i cin 3468    C_ wss 3469   class class class wbr 4440   Se wse 4829    We wwe 4830   dom cdm 4992    |` cres 4994   Rel wrel 4997   Fun wfun 5573    Fn wfn 5574   ` cfv 5579   Predcpred 28806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-fv 5587  df-pred 28807
This theorem is referenced by:  wfrlem11  28916
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