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Theorem frrlem4 24355
Description: Lemma for founded recursion. Properties of the restriction of an acceptable function to the domain of another acceptable function. (Contributed by Paul Chapman, 21-Apr-2012.)
Hypothesis
Ref Expression
frrlem4.1  |-  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
frrlem4  |-  ( ( 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 ) ) ) ) )
Distinct variable groups:    A, a,
f, g    A, h, x, y, a    B, a   
f, h, x, y    G, a, f, g    h, G, x, y    x, g, y    R, a, f, g    R, h, x, y
Allowed substitution hints:    B( x, y, f, g, h)

Proof of Theorem frrlem4
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frrlem4.1 . . . . . 6  |-  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 ) ) ) ) ) }
21frrlem2 24353 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
3 funfn 5299 . . . . 5  |-  ( Fun  g  <->  g  Fn  dom  g )
42, 3sylib 188 . . . 4  |-  ( g  e.  B  ->  g  Fn  dom  g )
5 fnresin1 5374 . . . 4  |-  ( g  Fn  dom  g  -> 
( g  |`  ( dom  g  i^i  dom  h
) )  Fn  ( dom  g  i^i  dom  h
) )
64, 5syl 15 . . 3  |-  ( g  e.  B  ->  (
g  |`  ( dom  g  i^i  dom  h ) )  Fn  ( dom  g  i^i  dom  h ) )
76adantr 451 . 2  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( g  |`  ( dom  g  i^i  dom  h
) )  Fn  ( dom  g  i^i  dom  h
) )
8 inss1 3402 . . . . . . . 8  |-  ( dom  g  i^i  dom  h
)  C_  dom  g
98sseli 3189 . . . . . . 7  |-  ( a  e.  ( dom  g  i^i  dom  h )  -> 
a  e.  dom  g
)
101frrlem1 24352 . . . . . . . . 9  |-  B  =  { g  |  E. b ( g  Fn  b  /\  ( b 
C_  A  /\  A. a  e.  b  Pred ( R ,  A , 
a )  C_  b  /\  A. a  e.  b  ( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) ) ) }
1110abeq2i 2403 . . . . . . . 8  |-  ( g  e.  B  <->  E. b
( g  Fn  b  /\  ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) ) )
12 fndm 5359 . . . . . . . . . 10  |-  ( g  Fn  b  ->  dom  g  =  b )
13 rsp 2616 . . . . . . . . . . . 12  |-  ( A. a  e.  b  (
g `  a )  =  ( a G ( g  |`  Pred ( R ,  A , 
a ) ) )  ->  ( a  e.  b  ->  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) )
14133ad2ant3 978 . . . . . . . . . . 11  |-  ( ( b  C_  A  /\  A. a  e.  b  Pred ( R ,  A , 
a )  C_  b  /\  A. a  e.  b  ( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) )  ->  (
a  e.  b  -> 
( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) ) )
15 eleq2 2357 . . . . . . . . . . . 12  |-  ( dom  g  =  b  -> 
( a  e.  dom  g 
<->  a  e.  b ) )
1615imbi1d 308 . . . . . . . . . . 11  |-  ( dom  g  =  b  -> 
( ( a  e. 
dom  g  ->  (
g `  a )  =  ( a G ( g  |`  Pred ( R ,  A , 
a ) ) ) )  <->  ( a  e.  b  ->  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) ) )
1714, 16syl5ibrcom 213 . . . . . . . . . 10  |-  ( ( b  C_  A  /\  A. a  e.  b  Pred ( R ,  A , 
a )  C_  b  /\  A. a  e.  b  ( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) )  ->  ( dom  g  =  b  ->  ( a  e.  dom  g  ->  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) ) )
1812, 17mpan9 455 . . . . . . . . 9  |-  ( ( g  Fn  b  /\  ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) )  ->  ( a  e. 
dom  g  ->  (
g `  a )  =  ( a G ( g  |`  Pred ( R ,  A , 
a ) ) ) ) )
1918exlimiv 1624 . . . . . . . 8  |-  ( E. b ( g  Fn  b  /\  ( b 
C_  A  /\  A. a  e.  b  Pred ( R ,  A , 
a )  C_  b  /\  A. a  e.  b  ( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) ) )  -> 
( a  e.  dom  g  ->  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) )
2011, 19sylbi 187 . . . . . . 7  |-  ( g  e.  B  ->  (
a  e.  dom  g  ->  ( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) ) )
219, 20syl5 28 . . . . . 6  |-  ( g  e.  B  ->  (
a  e.  ( dom  g  i^i  dom  h
)  ->  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) )
2221imp 418 . . . . 5  |-  ( ( g  e.  B  /\  a  e.  ( dom  g  i^i  dom  h )
)  ->  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) )
2322adantlr 695 . . . 4  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  a  e.  ( dom  g  i^i  dom  h ) )  -> 
( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) )
24 fvres 5558 . . . . 5  |-  ( a  e.  ( dom  g  i^i  dom  h )  -> 
( ( g  |`  ( dom  g  i^i  dom  h ) ) `  a )  =  ( g `  a ) )
2524adantl 452 . . . 4  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  a  e.  ( dom  g  i^i  dom  h ) )  -> 
( ( g  |`  ( dom  g  i^i  dom  h ) ) `  a )  =  ( g `  a ) )
26 resres 4984 . . . . . 6  |-  ( ( g  |`  ( dom  g  i^i  dom  h )
)  |`  Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) )  =  ( g  |`  ( ( dom  g  i^i  dom  h
)  i^i  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) )
27 predss 24244 . . . . . . . . 9  |-  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )  C_  ( dom  g  i^i  dom  h )
28 sseqin2 3401 . . . . . . . . 9  |-  ( Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )  C_  ( dom  g  i^i  dom  h )  <->  ( ( dom  g  i^i 
dom  h )  i^i 
Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) )  =  Pred ( R ,  ( dom  g  i^i  dom  h
) ,  a ) )
2927, 28mpbi 199 . . . . . . . 8  |-  ( ( dom  g  i^i  dom  h )  i^i  Pred ( R ,  ( dom  g  i^i  dom  h
) ,  a ) )  =  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )
301frrlem1 24352 . . . . . . . . . . . 12  |-  B  =  { h  |  E. c ( h  Fn  c  /\  ( c 
C_  A  /\  A. a  e.  c  Pred ( R ,  A , 
a )  C_  c  /\  A. a  e.  c  ( h `  a
)  =  ( a G ( h  |`  Pred ( R ,  A ,  a ) ) ) ) ) }
3130abeq2i 2403 . . . . . . . . . . 11  |-  ( h  e.  B  <->  E. c
( h  Fn  c  /\  ( c  C_  A  /\  A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c  /\  A. a  e.  c  ( h `  a )  =  ( a G ( h  |`  Pred ( R ,  A ,  a )
) ) ) ) )
32 eeanv 1866 . . . . . . . . . . . 12  |-  ( E. b E. c ( ( g  Fn  b  /\  ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) )  /\  ( h  Fn  c  /\  ( c 
C_  A  /\  A. a  e.  c  Pred ( R ,  A , 
a )  C_  c  /\  A. a  e.  c  ( h `  a
)  =  ( a G ( h  |`  Pred ( R ,  A ,  a ) ) ) ) ) )  <-> 
( E. b ( g  Fn  b  /\  ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) )  /\  E. c ( h  Fn  c  /\  ( c  C_  A  /\  A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c  /\  A. a  e.  c  ( h `  a )  =  ( a G ( h  |`  Pred ( R ,  A ,  a )
) ) ) ) ) )
33 simpl1 958 . . . . . . . . . . . . . . . . 17  |-  ( ( ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) )  /\  ( c  C_  A  /\  A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c  /\  A. a  e.  c  ( h `  a )  =  ( a G ( h  |`  Pred ( R ,  A ,  a )
) ) ) )  ->  b  C_  A
)
34 ssinss1 3410 . . . . . . . . . . . . . . . . 17  |-  ( b 
C_  A  ->  (
b  i^i  c )  C_  A )
3533, 34syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) )  /\  ( c  C_  A  /\  A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c  /\  A. a  e.  c  ( h `  a )  =  ( a G ( h  |`  Pred ( R ,  A ,  a )
) ) ) )  ->  ( b  i^i  c )  C_  A
)
36 simpl2 959 . . . . . . . . . . . . . . . . 17  |-  ( ( ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) )  /\  ( c  C_  A  /\  A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c  /\  A. a  e.  c  ( h `  a )  =  ( a G ( h  |`  Pred ( R ,  A ,  a )
) ) ) )  ->  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b )
37 simpr2 962 . . . . . . . . . . . . . . . . 17  |-  ( ( ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) )  /\  ( c  C_  A  /\  A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c  /\  A. a  e.  c  ( h `  a )  =  ( a G ( h  |`  Pred ( R ,  A ,  a )
) ) ) )  ->  A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c )
38 nfra1 2606 . . . . . . . . . . . . . . . . . . 19  |-  F/ a A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b
39 nfra1 2606 . . . . . . . . . . . . . . . . . . 19  |-  F/ a A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c
4038, 39nfan 1783 . . . . . . . . . . . . . . . . . 18  |-  F/ a ( A. a  e.  b  Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  c  Pred ( R ,  A ,  a )  C_  c )
41 inss1 3402 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  i^i  c )  C_  b
4241sseli 3189 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  e.  ( b  i^i  c )  ->  a  e.  b )
43 rsp 2616 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A. a  e.  b  Pred ( R ,  A , 
a )  C_  b  ->  ( a  e.  b  ->  Pred ( R ,  A ,  a )  C_  b ) )
4442, 43syl5com 26 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  e.  ( b  i^i  c )  ->  ( A. a  e.  b  Pred ( R ,  A ,  a )  C_  b  ->  Pred ( R ,  A ,  a )  C_  b ) )
45 inss2 3403 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  i^i  c )  C_  c
4645sseli 3189 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  e.  ( b  i^i  c )  ->  a  e.  c )
47 rsp 2616 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A. a  e.  c  Pred ( R ,  A , 
a )  C_  c  ->  ( a  e.  c  ->  Pred ( R ,  A ,  a )  C_  c ) )
4846, 47syl5com 26 . . . . . . . . . . . . . . . . . . . 20  |-  ( a  e.  ( b  i^i  c )  ->  ( A. a  e.  c  Pred ( R ,  A ,  a )  C_  c  ->  Pred ( R ,  A ,  a )  C_  c ) )
4944, 48anim12d 546 . . . . . . . . . . . . . . . . . . 19  |-  ( a  e.  ( b  i^i  c )  ->  (
( A. a  e.  b  Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  c  Pred ( R ,  A ,  a )  C_  c )  ->  ( Pred ( R ,  A ,  a )  C_  b  /\  Pred ( R ,  A ,  a )  C_  c ) ) )
50 ssin 3404 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
Pred ( R ,  A ,  a )  C_  b  /\  Pred ( R ,  A , 
a )  C_  c
)  <->  Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) )
5150biimpi 186 . . . . . . . . . . . . . . . . . . 19  |-  ( (
Pred ( R ,  A ,  a )  C_  b  /\  Pred ( R ,  A , 
a )  C_  c
)  ->  Pred ( R ,  A ,  a )  C_  ( b  i^i  c ) )
5249, 51syl6com 31 . . . . . . . . . . . . . . . . . 18  |-  ( ( A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  c  Pred ( R ,  A ,  a )  C_  c )  ->  ( a  e.  ( b  i^i  c )  ->  Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) ) )
5340, 52ralrimi 2637 . . . . . . . . . . . . . . . . 17  |-  ( ( A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  c  Pred ( R ,  A ,  a )  C_  c )  ->  A. a  e.  ( b  i^i  c )
Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) )
5436, 37, 53syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) )  /\  ( c  C_  A  /\  A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c  /\  A. a  e.  c  ( h `  a )  =  ( a G ( h  |`  Pred ( R ,  A ,  a )
) ) ) )  ->  A. a  e.  ( b  i^i  c )
Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) )
55 fndm 5359 . . . . . . . . . . . . . . . . . 18  |-  ( h  Fn  c  ->  dom  h  =  c )
56 ineq12 3378 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( dom  g  =  b  /\  dom  h  =  c )  ->  ( dom  g  i^i  dom  h
)  =  ( b  i^i  c ) )
5756sseq1d 3218 . . . . . . . . . . . . . . . . . . 19  |-  ( ( dom  g  =  b  /\  dom  h  =  c )  ->  (
( dom  g  i^i  dom  h )  C_  A  <->  ( b  i^i  c ) 
C_  A ) )
58 sseq2 3213 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( dom  g  i^i  dom  h )  =  ( b  i^i  c )  ->  ( Pred ( R ,  A , 
a )  C_  ( dom  g  i^i  dom  h
)  <->  Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) ) )
5958raleqbi1dv 2757 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( dom  g  i^i  dom  h )  =  ( b  i^i  c )  ->  ( A. a  e.  ( dom  g  i^i 
dom  h ) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i  dom  h )  <->  A. a  e.  ( b  i^i  c
) Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) ) )
6056, 59syl 15 . . . . . . . . . . . . . . . . . . 19  |-  ( ( dom  g  =  b  /\  dom  h  =  c )  ->  ( A. a  e.  ( dom  g  i^i  dom  h
) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i 
dom  h )  <->  A. a  e.  ( b  i^i  c
) Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) ) )
6157, 60anbi12d 691 . . . . . . . . . . . . . . . . . 18  |-  ( ( dom  g  =  b  /\  dom  h  =  c )  ->  (
( ( dom  g  i^i  dom  h )  C_  A  /\  A. a  e.  ( dom  g  i^i 
dom  h ) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i  dom  h ) )  <->  ( (
b  i^i  c )  C_  A  /\  A. a  e.  ( b  i^i  c
) Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) ) ) )
6212, 55, 61syl2an 463 . . . . . . . . . . . . . . . . 17  |-  ( ( g  Fn  b  /\  h  Fn  c )  ->  ( ( ( dom  g  i^i  dom  h
)  C_  A  /\  A. a  e.  ( dom  g  i^i  dom  h
) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i 
dom  h ) )  <-> 
( ( b  i^i  c )  C_  A  /\  A. a  e.  ( b  i^i  c )
Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) ) ) )
6362biimprcd 216 . . . . . . . . . . . . . . . 16  |-  ( ( ( b  i^i  c
)  C_  A  /\  A. a  e.  ( b  i^i  c ) Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) )  ->  (
( g  Fn  b  /\  h  Fn  c
)  ->  ( ( dom  g  i^i  dom  h
)  C_  A  /\  A. a  e.  ( dom  g  i^i  dom  h
) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i 
dom  h ) ) ) )
6435, 54, 63syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) )  /\  ( c  C_  A  /\  A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c  /\  A. a  e.  c  ( h `  a )  =  ( a G ( h  |`  Pred ( R ,  A ,  a )
) ) ) )  ->  ( ( g  Fn  b  /\  h  Fn  c )  ->  (
( dom  g  i^i  dom  h )  C_  A  /\  A. a  e.  ( dom  g  i^i  dom  h ) Pred ( R ,  A , 
a )  C_  ( dom  g  i^i  dom  h
) ) ) )
6564impcom 419 . . . . . . . . . . . . . 14  |-  ( ( ( g  Fn  b  /\  h  Fn  c
)  /\  ( (
b  C_  A  /\  A. a  e.  b  Pred ( R ,  A , 
a )  C_  b  /\  A. a  e.  b  ( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) )  /\  (
c  C_  A  /\  A. a  e.  c  Pred ( R ,  A , 
a )  C_  c  /\  A. a  e.  c  ( h `  a
)  =  ( a G ( h  |`  Pred ( R ,  A ,  a ) ) ) ) ) )  ->  ( ( dom  g  i^i  dom  h
)  C_  A  /\  A. a  e.  ( dom  g  i^i  dom  h
) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i 
dom  h ) ) )
6665an4s 799 . . . . . . . . . . . . 13  |-  ( ( ( g  Fn  b  /\  ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) )  /\  ( h  Fn  c  /\  ( c 
C_  A  /\  A. a  e.  c  Pred ( R ,  A , 
a )  C_  c  /\  A. a  e.  c  ( h `  a
)  =  ( a G ( h  |`  Pred ( R ,  A ,  a ) ) ) ) ) )  ->  ( ( dom  g  i^i  dom  h
)  C_  A  /\  A. a  e.  ( dom  g  i^i  dom  h
) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i 
dom  h ) ) )
6766exlimivv 1625 . . . . . . . . . . . 12  |-  ( E. b E. c ( ( g  Fn  b  /\  ( b  C_  A  /\  A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b  /\  A. a  e.  b  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) )  /\  ( h  Fn  c  /\  ( c 
C_  A  /\  A. a  e.  c  Pred ( R ,  A , 
a )  C_  c  /\  A. a  e.  c  ( h `  a
)  =  ( a G ( h  |`  Pred ( R ,  A ,  a ) ) ) ) ) )  ->  ( ( dom  g  i^i  dom  h
)  C_  A  /\  A. a  e.  ( dom  g  i^i  dom  h
) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i 
dom  h ) ) )
6832, 67sylbir 204 . . . . . . . . . . 11  |-  ( ( E. b ( g  Fn  b  /\  (
b  C_  A  /\  A. a  e.  b  Pred ( R ,  A , 
a )  C_  b  /\  A. a  e.  b  ( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) ) )  /\  E. c ( h  Fn  c  /\  ( c 
C_  A  /\  A. a  e.  c  Pred ( R ,  A , 
a )  C_  c  /\  A. a  e.  c  ( h `  a
)  =  ( a G ( h  |`  Pred ( R ,  A ,  a ) ) ) ) ) )  ->  ( ( dom  g  i^i  dom  h
)  C_  A  /\  A. a  e.  ( dom  g  i^i  dom  h
) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i 
dom  h ) ) )
6911, 31, 68syl2anb 465 . . . . . . . . . 10  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( dom  g  i^i  dom  h )  C_  A  /\  A. a  e.  ( dom  g  i^i 
dom  h ) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i  dom  h ) ) )
7069adantr 451 . . . . . . . . 9  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  a  e.  ( dom  g  i^i  dom  h ) )  -> 
( ( dom  g  i^i  dom  h )  C_  A  /\  A. a  e.  ( dom  g  i^i 
dom  h ) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i  dom  h ) ) )
71 simpr 447 . . . . . . . . 9  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  a  e.  ( dom  g  i^i  dom  h ) )  -> 
a  e.  ( dom  g  i^i  dom  h
) )
72 preddowncl 24267 . . . . . . . . 9  |-  ( ( ( dom  g  i^i 
dom  h )  C_  A  /\  A. a  e.  ( dom  g  i^i 
dom  h ) Pred ( R ,  A ,  a )  C_  ( dom  g  i^i  dom  h ) )  -> 
( a  e.  ( dom  g  i^i  dom  h )  ->  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )  = 
Pred ( R ,  A ,  a )
) )
7370, 71, 72sylc 56 . . . . . . . 8  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  a  e.  ( dom  g  i^i  dom  h ) )  ->  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )  =  Pred ( R ,  A , 
a ) )
7429, 73syl5eq 2340 . . . . . . 7  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  a  e.  ( dom  g  i^i  dom  h ) )  -> 
( ( dom  g  i^i  dom  h )  i^i 
Pred ( R , 
( dom  g  i^i  dom  h ) ,  a ) )  =  Pred ( R ,  A , 
a ) )
7574reseq2d 4971 . . . . . 6  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  a  e.  ( dom  g  i^i  dom  h ) )  -> 
( g  |`  (
( dom  g  i^i  dom  h )  i^i  Pred ( R ,  ( dom  g  i^i  dom  h
) ,  a ) ) )  =  ( g  |`  Pred ( R ,  A ,  a ) ) )
7626, 75syl5eq 2340 . . . . 5  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  a  e.  ( dom  g  i^i  dom  h ) )  -> 
( ( g  |`  ( dom  g  i^i  dom  h ) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) )  =  ( g  |`  Pred ( R ,  A ,  a ) ) )
7776oveq2d 5890 . . . 4  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  a  e.  ( dom  g  i^i  dom  h ) )  -> 
( a G ( ( g  |`  ( dom  g  i^i  dom  h
) )  |`  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a ) ) )  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) )
7823, 25, 773eqtr4d 2338 . . 3  |-  ( ( ( g  e.  B  /\  h  e.  B
)  /\  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 ) ) ) )
7978ralrimiva 2639 . 2  |-  ( ( g  e.  B  /\  h  e.  B )  ->  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 ) ) ) )
807, 79jca 518 1  |-  ( ( 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 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556    i^i cin 3164    C_ wss 3165   dom cdm 4705    |` cres 4707   Fun wfun 5265    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   Predcpred 24238
This theorem is referenced by:  frrlem5  24356
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  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-fv 5279  df-ov 5877  df-pred 24239
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