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Theorem frrlem4 25577
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 25575 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
3 funfn 5474 . . . . 5  |-  ( Fun  g  <->  g  Fn  dom  g )
42, 3sylib 189 . . . 4  |-  ( g  e.  B  ->  g  Fn  dom  g )
5 fnresin1 5551 . . . 4  |-  ( g  Fn  dom  g  -> 
( g  |`  ( dom  g  i^i  dom  h
) )  Fn  ( dom  g  i^i  dom  h
) )
64, 5syl 16 . . 3  |-  ( g  e.  B  ->  (
g  |`  ( dom  g  i^i  dom  h ) )  Fn  ( dom  g  i^i  dom  h ) )
76adantr 452 . 2  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( g  |`  ( dom  g  i^i  dom  h
) )  Fn  ( dom  g  i^i  dom  h
) )
8 inss1 3553 . . . . . . . 8  |-  ( dom  g  i^i  dom  h
)  C_  dom  g
98sseli 3336 . . . . . . 7  |-  ( a  e.  ( dom  g  i^i  dom  h )  -> 
a  e.  dom  g
)
101frrlem1 25574 . . . . . . . . 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 2542 . . . . . . . 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 5536 . . . . . . . . . 10  |-  ( g  Fn  b  ->  dom  g  =  b )
13 rsp 2758 . . . . . . . . . . . 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 980 . . . . . . . . . . 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 2496 . . . . . . . . . . . 12  |-  ( dom  g  =  b  -> 
( a  e.  dom  g 
<->  a  e.  b ) )
1615imbi1d 309 . . . . . . . . . . 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 214 . . . . . . . . . 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 456 . . . . . . . . 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 1644 . . . . . . . 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 188 . . . . . . 7  |-  ( g  e.  B  ->  (
a  e.  dom  g  ->  ( g `  a
)  =  ( a G ( g  |`  Pred ( R ,  A ,  a ) ) ) ) )
219, 20syl5 30 . . . . . 6  |-  ( g  e.  B  ->  (
a  e.  ( dom  g  i^i  dom  h
)  ->  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) ) )
2221imp 419 . . . . 5  |-  ( ( g  e.  B  /\  a  e.  ( dom  g  i^i  dom  h )
)  ->  ( g `  a )  =  ( a G ( g  |`  Pred ( R ,  A ,  a )
) ) )
2322adantlr 696 . . . 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 5737 . . . . 5  |-  ( a  e.  ( dom  g  i^i  dom  h )  -> 
( ( g  |`  ( dom  g  i^i  dom  h ) ) `  a )  =  ( g `  a ) )
2524adantl 453 . . . 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 5151 . . . . . 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 25438 . . . . . . . . 9  |-  Pred ( R ,  ( dom  g  i^i  dom  h ) ,  a )  C_  ( dom  g  i^i  dom  h )
28 sseqin2 3552 . . . . . . . . 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 200 . . . . . . . 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 25574 . . . . . . . . . . . 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 2542 . . . . . . . . . . 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 1937 . . . . . . . . . . . 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 960 . . . . . . . . . . . . . . . . 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 3561 . . . . . . . . . . . . . . . . 17  |-  ( b 
C_  A  ->  (
b  i^i  c )  C_  A )
3533, 34syl 16 . . . . . . . . . . . . . . . 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 961 . . . . . . . . . . . . . . . . 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 964 . . . . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . . . . . . . . 19  |-  F/ a A. a  e.  b 
Pred ( R ,  A ,  a )  C_  b
39 nfra1 2748 . . . . . . . . . . . . . . . . . . 19  |-  F/ a A. a  e.  c 
Pred ( R ,  A ,  a )  C_  c
4038, 39nfan 1846 . . . . . . . . . . . . . . . . . 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 3553 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  i^i  c )  C_  b
4241sseli 3336 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  e.  ( b  i^i  c )  ->  a  e.  b )
43 rsp 2758 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A. a  e.  b  Pred ( R ,  A , 
a )  C_  b  ->  ( a  e.  b  ->  Pred ( R ,  A ,  a )  C_  b ) )
4442, 43syl5com 28 . . . . . . . . . . . . . . . . . . . 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 3554 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( b  i^i  c )  C_  c
4645sseli 3336 . . . . . . . . . . . . . . . . . . . . 21  |-  ( a  e.  ( b  i^i  c )  ->  a  e.  c )
47 rsp 2758 . . . . . . . . . . . . . . . . . . . . 21  |-  ( A. a  e.  c  Pred ( R ,  A , 
a )  C_  c  ->  ( a  e.  c  ->  Pred ( R ,  A ,  a )  C_  c ) )
4846, 47syl5com 28 . . . . . . . . . . . . . . . . . . . 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 547 . . . . . . . . . . . . . . . . . . 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 3555 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
Pred ( R ,  A ,  a )  C_  b  /\  Pred ( R ,  A , 
a )  C_  c
)  <->  Pred ( R ,  A ,  a )  C_  ( b  i^i  c
) )
5150biimpi 187 . . . . . . . . . . . . . . . . . . 19  |-  ( (
Pred ( R ,  A ,  a )  C_  b  /\  Pred ( R ,  A , 
a )  C_  c
)  ->  Pred ( R ,  A ,  a )  C_  ( b  i^i  c ) )
5249, 51syl6com 33 . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . 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 5536 . . . . . . . . . . . . . . . . . 18  |-  ( h  Fn  c  ->  dom  h  =  c )
56 ineq12 3529 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( dom  g  =  b  /\  dom  h  =  c )  ->  ( dom  g  i^i  dom  h
)  =  ( b  i^i  c ) )
5756sseq1d 3367 . . . . . . . . . . . . . . . . . . 19  |-  ( ( dom  g  =  b  /\  dom  h  =  c )  ->  (
( dom  g  i^i  dom  h )  C_  A  <->  ( b  i^i  c ) 
C_  A ) )
58 sseq2 3362 . . . . . . . . . . . . . . . . . . . . 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 2904 . . . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . 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 1645 . . . . . . . . . . . 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 205 . . . . . . . . . . 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 466 . . . . . . . . . 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 452 . . . . . . . . 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 448 . . . . . . . . 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 25463 . . . . . . . . 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 58 . . . . . . . 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 2479 . . . . . . 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 5138 . . . . . 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 2479 . . . . 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 6089 . . . 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 2477 . . 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 2781 . 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 519 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 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697    i^i cin 3311    C_ wss 3312   dom cdm 4870    |` cres 4872   Fun wfun 5440    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   Predcpred 25430
This theorem is referenced by:  frrlem5  25578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-ov 6076  df-pred 25431
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