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Theorem erdszelem9 23745
Description: Lemma for erdsze 23748. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze.n  |-  ( ph  ->  N  e.  NN )
erdsze.f  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
erdszelem.i  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.j  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
erdszelem.t  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
Assertion
Ref Expression
erdszelem9  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
Distinct variable groups:    x, y, n, F    n, I, x, y    n, J, x, y    n, N, x, y    ph, n, x, y
Allowed substitution hints:    T( x, y, n)

Proof of Theorem erdszelem9
Dummy variables  w  z  a  b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 erdsze.n . . . . . 6  |-  ( ph  ->  N  e.  NN )
2 erdsze.f . . . . . 6  |-  ( ph  ->  F : ( 1 ... N ) -1-1-> RR )
3 erdszelem.i . . . . . 6  |-  I  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  <  (
y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
4 ltso 8919 . . . . . 6  |-  <  Or  RR
51, 2, 3, 4erdszelem6 23742 . . . . 5  |-  ( ph  ->  I : ( 1 ... N ) --> NN )
6 ffvelrn 5679 . . . . 5  |-  ( ( I : ( 1 ... N ) --> NN 
/\  n  e.  ( 1 ... N ) )  ->  ( I `  n )  e.  NN )
75, 6sylan 457 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
I `  n )  e.  NN )
8 erdszelem.j . . . . . 6  |-  J  =  ( x  e.  ( 1 ... N ) 
|->  sup ( ( # " { y  e.  ~P ( 1 ... x
)  |  ( ( F  |`  y )  Isom  <  ,  `'  <  ( y ,  ( F
" y ) )  /\  x  e.  y ) } ) ,  RR ,  <  )
)
9 cnvso 5230 . . . . . . 7  |-  (  < 
Or  RR  <->  `'  <  Or  RR )
104, 9mpbi 199 . . . . . 6  |-  `'  <  Or  RR
111, 2, 8, 10erdszelem6 23742 . . . . 5  |-  ( ph  ->  J : ( 1 ... N ) --> NN )
12 ffvelrn 5679 . . . . 5  |-  ( ( J : ( 1 ... N ) --> NN 
/\  n  e.  ( 1 ... N ) )  ->  ( J `  n )  e.  NN )
1311, 12sylan 457 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  ( J `  n )  e.  NN )
14 opelxpi 4737 . . . 4  |-  ( ( ( I `  n
)  e.  NN  /\  ( J `  n )  e.  NN )  ->  <. ( I `  n
) ,  ( J `
 n ) >.  e.  ( NN  X.  NN ) )
157, 13, 14syl2anc 642 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  <. (
I `  n ) ,  ( J `  n ) >.  e.  ( NN  X.  NN ) )
16 erdszelem.t . . 3  |-  T  =  ( n  e.  ( 1 ... N ) 
|->  <. ( I `  n ) ,  ( J `  n )
>. )
1715, 16fmptd 5700 . 2  |-  ( ph  ->  T : ( 1 ... N ) --> ( NN  X.  NN ) )
18 fveq2 5541 . . . . . 6  |-  ( a  =  z  ->  ( T `  a )  =  ( T `  z ) )
19 fveq2 5541 . . . . . 6  |-  ( b  =  w  ->  ( T `  b )  =  ( T `  w ) )
2018, 19eqeqan12d 2311 . . . . 5  |-  ( ( a  =  z  /\  b  =  w )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  z
)  =  ( T `
 w ) ) )
21 eqeq12 2308 . . . . 5  |-  ( ( a  =  z  /\  b  =  w )  ->  ( a  =  b  <-> 
z  =  w ) )
2220, 21imbi12d 311 . . . 4  |-  ( ( a  =  z  /\  b  =  w )  ->  ( ( ( T `
 a )  =  ( T `  b
)  ->  a  =  b )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
23 fveq2 5541 . . . . . . 7  |-  ( a  =  w  ->  ( T `  a )  =  ( T `  w ) )
24 fveq2 5541 . . . . . . 7  |-  ( b  =  z  ->  ( T `  b )  =  ( T `  z ) )
2523, 24eqeqan12d 2311 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  w
)  =  ( T `
 z ) ) )
26 eqcom 2298 . . . . . 6  |-  ( ( T `  w )  =  ( T `  z )  <->  ( T `  z )  =  ( T `  w ) )
2725, 26syl6bb 252 . . . . 5  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( T `  a )  =  ( T `  b )  <-> 
( T `  z
)  =  ( T `
 w ) ) )
28 eqeq12 2308 . . . . . 6  |-  ( ( a  =  w  /\  b  =  z )  ->  ( a  =  b  <-> 
w  =  z ) )
29 eqcom 2298 . . . . . 6  |-  ( w  =  z  <->  z  =  w )
3028, 29syl6bb 252 . . . . 5  |-  ( ( a  =  w  /\  b  =  z )  ->  ( a  =  b  <-> 
z  =  w ) )
3127, 30imbi12d 311 . . . 4  |-  ( ( a  =  w  /\  b  =  z )  ->  ( ( ( T `
 a )  =  ( T `  b
)  ->  a  =  b )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
32 elfzelz 10814 . . . . . . 7  |-  ( z  e.  ( 1 ... N )  ->  z  e.  ZZ )
3332zred 10133 . . . . . 6  |-  ( z  e.  ( 1 ... N )  ->  z  e.  RR )
3433ssriv 3197 . . . . 5  |-  ( 1 ... N )  C_  RR
3534a1i 10 . . . 4  |-  ( ph  ->  ( 1 ... N
)  C_  RR )
36 biidd 228 . . . 4  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
) ) )  -> 
( ( ( T `
 z )  =  ( T `  w
)  ->  z  =  w )  <->  ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) ) )
37 simpr1 961 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  e.  ( 1 ... N
) )
38 fveq2 5541 . . . . . . . . . 10  |-  ( n  =  z  ->  (
I `  n )  =  ( I `  z ) )
39 fveq2 5541 . . . . . . . . . 10  |-  ( n  =  z  ->  ( J `  n )  =  ( J `  z ) )
4038, 39opeq12d 3820 . . . . . . . . 9  |-  ( n  =  z  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  z ) ,  ( J `  z ) >. )
41 opex 4253 . . . . . . . . 9  |-  <. (
I `  z ) ,  ( J `  z ) >.  e.  _V
4240, 16, 41fvmpt 5618 . . . . . . . 8  |-  ( z  e.  ( 1 ... N )  ->  ( T `  z )  =  <. ( I `  z ) ,  ( J `  z )
>. )
4337, 42syl 15 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( T `  z )  =  <. ( I `  z ) ,  ( J `  z )
>. )
44 simpr2 962 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  w  e.  ( 1 ... N
) )
45 fveq2 5541 . . . . . . . . . 10  |-  ( n  =  w  ->  (
I `  n )  =  ( I `  w ) )
46 fveq2 5541 . . . . . . . . . 10  |-  ( n  =  w  ->  ( J `  n )  =  ( J `  w ) )
4745, 46opeq12d 3820 . . . . . . . . 9  |-  ( n  =  w  ->  <. (
I `  n ) ,  ( J `  n ) >.  =  <. ( I `  w ) ,  ( J `  w ) >. )
48 opex 4253 . . . . . . . . 9  |-  <. (
I `  w ) ,  ( J `  w ) >.  e.  _V
4947, 16, 48fvmpt 5618 . . . . . . . 8  |-  ( w  e.  ( 1 ... N )  ->  ( T `  w )  =  <. ( I `  w ) ,  ( J `  w )
>. )
5044, 49syl 15 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( T `  w )  =  <. ( I `  w ) ,  ( J `  w )
>. )
5143, 50eqeq12d 2310 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  <->  <. ( I `
 z ) ,  ( J `  z
) >.  =  <. (
I `  w ) ,  ( J `  w ) >. )
)
52 fvex 5555 . . . . . . . 8  |-  ( I `
 z )  e. 
_V
53 fvex 5555 . . . . . . . 8  |-  ( J `
 z )  e. 
_V
5452, 53opth 4261 . . . . . . 7  |-  ( <.
( I `  z
) ,  ( J `
 z ) >.  =  <. ( I `  w ) ,  ( J `  w )
>. 
<->  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) )
5537, 33syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  e.  RR )
5634, 44sseldi 3191 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  w  e.  RR )
57 simpr3 963 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  z  <_  w )
5855, 56, 57leltned 8986 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  <->  w  =/=  z ) )
592adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  F : ( 1 ... N ) -1-1-> RR )
60 f1fveq 5802 . . . . . . . . . . . . . . . . 17  |-  ( ( F : ( 1 ... N ) -1-1-> RR  /\  ( z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N ) ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  z  =  w ) )
6159, 37, 44, 60syl12anc 1180 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  z  =  w ) )
6261, 29syl6bbr 254 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =  ( F `
 w )  <->  w  =  z ) )
6362necon3bid 2494 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( F `  z
)  =/=  ( F `
 w )  <->  w  =/=  z ) )
6458, 63bitr4d 247 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  <->  ( F `  z )  =/=  ( F `  w )
) )
6564biimpa 470 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  z )  =/=  ( F `  w )
)
66 f1f 5453 . . . . . . . . . . . . . . . 16  |-  ( F : ( 1 ... N ) -1-1-> RR  ->  F : ( 1 ... N ) --> RR )
672, 66syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( 1 ... N ) --> RR )
6867ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  F :
( 1 ... N
) --> RR )
6937adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  z  e.  ( 1 ... N
) )
70 ffvelrn 5679 . . . . . . . . . . . . . 14  |-  ( ( F : ( 1 ... N ) --> RR 
/\  z  e.  ( 1 ... N ) )  ->  ( F `  z )  e.  RR )
7168, 69, 70syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  z )  e.  RR )
7244adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  w  e.  ( 1 ... N
) )
73 ffvelrn 5679 . . . . . . . . . . . . . 14  |-  ( ( F : ( 1 ... N ) --> RR 
/\  w  e.  ( 1 ... N ) )  ->  ( F `  w )  e.  RR )
7468, 72, 73syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( F `  w )  e.  RR )
7571, 74lttri2d 8974 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( F `  z )  =/=  ( F `  w
)  <->  ( ( F `
 z )  < 
( F `  w
)  \/  ( F `
 w )  < 
( F `  z
) ) ) )
7665, 75mpbid 201 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( F `  z )  <  ( F `  w
)  \/  ( F `
 w )  < 
( F `  z
) ) )
771ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  N  e.  NN )
782ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  F :
( 1 ... N
) -1-1-> RR )
79 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  z  <  w )
8077, 78, 3, 4, 69, 72, 79erdszelem8 23744 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
I `  z )  =  ( I `  w )  ->  -.  ( F `  z )  <  ( F `  w ) ) )
8177, 78, 8, 10, 69, 72, 79erdszelem8 23744 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( ( J `  z )  =  ( J `  w )  ->  -.  ( F `  z ) `'  <  ( F `  w ) ) )
8280, 81anim12d 546 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) )  -> 
( -.  ( F `
 z )  < 
( F `  w
)  /\  -.  ( F `  z ) `'  <  ( F `  w ) ) ) )
83 ioran 476 . . . . . . . . . . . . 13  |-  ( -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) )  <->  ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  w )  <  ( F `  z ) ) )
84 fvex 5555 . . . . . . . . . . . . . . . 16  |-  ( F `
 z )  e. 
_V
85 fvex 5555 . . . . . . . . . . . . . . . 16  |-  ( F `
 w )  e. 
_V
8684, 85brcnv 4880 . . . . . . . . . . . . . . 15  |-  ( ( F `  z ) `'  <  ( F `  w )  <->  ( F `  w )  <  ( F `  z )
)
8786notbii 287 . . . . . . . . . . . . . 14  |-  ( -.  ( F `  z
) `'  <  ( F `  w )  <->  -.  ( F `  w
)  <  ( F `  z ) )
8887anbi2i 675 . . . . . . . . . . . . 13  |-  ( ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  z ) `'  <  ( F `  w ) )  <->  ( -.  ( F `  z )  <  ( F `  w
)  /\  -.  ( F `  w )  <  ( F `  z
) ) )
8983, 88bitr4i 243 . . . . . . . . . . . 12  |-  ( -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) )  <->  ( -.  ( F `  z )  <  ( F `  w )  /\  -.  ( F `  z ) `'  <  ( F `  w ) ) )
9082, 89syl6ibr 218 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  ( (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) )  ->  -.  ( ( F `  z )  <  ( F `  w )  \/  ( F `  w
)  <  ( F `  z ) ) ) )
9176, 90mt2d 109 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
z  e.  ( 1 ... N )  /\  w  e.  ( 1 ... N )  /\  z  <_  w ) )  /\  z  <  w
)  ->  -.  (
( I `  z
)  =  ( I `
 w )  /\  ( J `  z )  =  ( J `  w ) ) )
9291ex 423 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
z  <  w  ->  -.  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) ) )
9358, 92sylbird 226 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
w  =/=  z  ->  -.  ( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) ) ) )
9493necon4ad 2520 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( ( I `  z )  =  ( I `  w )  /\  ( J `  z )  =  ( J `  w ) )  ->  w  =  z ) )
9554, 94syl5bi 208 . . . . . 6  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  ( <. ( I `  z
) ,  ( J `
 z ) >.  =  <. ( I `  w ) ,  ( J `  w )
>.  ->  w  =  z ) )
9651, 95sylbid 206 . . . . 5  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  ->  w  =  z )
)
9796, 29syl6ib 217 . . . 4  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
)  /\  z  <_  w ) )  ->  (
( T `  z
)  =  ( T `
 w )  -> 
z  =  w ) )
9822, 31, 35, 36, 97wlogle 9322 . . 3  |-  ( (
ph  /\  ( z  e.  ( 1 ... N
)  /\  w  e.  ( 1 ... N
) ) )  -> 
( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
9998ralrimivva 2648 . 2  |-  ( ph  ->  A. z  e.  ( 1 ... N ) A. w  e.  ( 1 ... N ) ( ( T `  z )  =  ( T `  w )  ->  z  =  w ) )
100 dff13 5799 . 2  |-  ( T : ( 1 ... N ) -1-1-> ( NN 
X.  NN )  <->  ( T : ( 1 ... N ) --> ( NN 
X.  NN )  /\  A. z  e.  ( 1 ... N ) A. w  e.  ( 1 ... N ) ( ( T `  z
)  =  ( T `
 w )  -> 
z  =  w ) ) )
10117, 99, 100sylanbrc 645 1  |-  ( ph  ->  T : ( 1 ... N ) -1-1-> ( NN  X.  NN ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560    C_ wss 3165   ~Pcpw 3638   <.cop 3656   class class class wbr 4039    e. cmpt 4093    Or wor 4329    X. cxp 4703   `'ccnv 4704    |` cres 4707   "cima 4708   -->wf 5267   -1-1->wf1 5268   ` cfv 5271    Isom wiso 5272  (class class class)co 5874   supcsup 7209   RRcr 8752   1c1 8754    < clt 8883    <_ cle 8884   NNcn 9762   ...cfz 10798   #chash 11353
This theorem is referenced by:  erdszelem10  23746
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354
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