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Theorem erdsze2lem1 28911
Description: Lemma for erdsze2 28913. (Contributed by Mario Carneiro, 22-Jan-2015.)
Hypotheses
Ref Expression
erdsze2.r  |-  ( ph  ->  R  e.  NN )
erdsze2.s  |-  ( ph  ->  S  e.  NN )
erdsze2.f  |-  ( ph  ->  F : A -1-1-> RR )
erdsze2.a  |-  ( ph  ->  A  C_  RR )
erdsze2lem.n  |-  N  =  ( ( R  - 
1 )  x.  ( S  -  1 ) )
erdsze2lem.l  |-  ( ph  ->  N  <  ( # `  A ) )
Assertion
Ref Expression
erdsze2lem1  |-  ( ph  ->  E. f ( f : ( 1 ... ( N  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
Distinct variable groups:    A, f    f, F    R, f    S, f   
f, N    ph, f

Proof of Theorem erdsze2lem1
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 erdsze2lem.n . . . . . . . . 9  |-  N  =  ( ( R  - 
1 )  x.  ( S  -  1 ) )
2 erdsze2.r . . . . . . . . . . 11  |-  ( ph  ->  R  e.  NN )
3 nnm1nn0 10833 . . . . . . . . . . 11  |-  ( R  e.  NN  ->  ( R  -  1 )  e.  NN0 )
42, 3syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( R  -  1 )  e.  NN0 )
5 erdsze2.s . . . . . . . . . . 11  |-  ( ph  ->  S  e.  NN )
6 nnm1nn0 10833 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
75, 6syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( S  -  1 )  e.  NN0 )
84, 7nn0mulcld 10853 . . . . . . . . 9  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  e.  NN0 )
91, 8syl5eqel 2546 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
10 peano2nn0 10832 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
11 hashfz1 12401 . . . . . . . 8  |-  ( ( N  +  1 )  e.  NN0  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
129, 10, 113syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( N  + 
1 ) )
1312adantr 463 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
14 erdsze2lem.l . . . . . . . 8  |-  ( ph  ->  N  <  ( # `  A ) )
1514adantr 463 . . . . . . 7  |-  ( (
ph  /\  A  e.  Fin )  ->  N  < 
( # `  A ) )
16 hashcl 12410 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
17 nn0ltp1le 10917 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( # `  A )  e.  NN0 )  -> 
( N  <  ( # `
 A )  <->  ( N  +  1 )  <_ 
( # `  A ) ) )
189, 16, 17syl2an 475 . . . . . . 7  |-  ( (
ph  /\  A  e.  Fin )  ->  ( N  <  ( # `  A
)  <->  ( N  + 
1 )  <_  ( # `
 A ) ) )
1915, 18mpbid 210 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( N  +  1 )  <_ 
( # `  A ) )
2013, 19eqbrtrd 4459 . . . . 5  |-  ( (
ph  /\  A  e.  Fin )  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  <_  ( # `
 A ) )
21 fzfid 12065 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( 1 ... ( N  + 
1 ) )  e. 
Fin )
22 simpr 459 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  A  e. 
Fin )
23 hashdom 12430 . . . . . 6  |-  ( ( ( 1 ... ( N  +  1 ) )  e.  Fin  /\  A  e.  Fin )  ->  ( ( # `  (
1 ... ( N  + 
1 ) ) )  <_  ( # `  A
)  <->  ( 1 ... ( N  +  1 ) )  ~<_  A ) )
2421, 22, 23syl2anc 659 . . . . 5  |-  ( (
ph  /\  A  e.  Fin )  ->  ( (
# `  ( 1 ... ( N  +  1 ) ) )  <_ 
( # `  A )  <-> 
( 1 ... ( N  +  1 ) )  ~<_  A ) )
2520, 24mpbid 210 . . . 4  |-  ( (
ph  /\  A  e.  Fin )  ->  ( 1 ... ( N  + 
1 ) )  ~<_  A )
26 simpr 459 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  -.  A  e.  Fin )
27 fzfid 12065 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
1 ... ( N  + 
1 ) )  e. 
Fin )
28 isinffi 8364 . . . . . 6  |-  ( ( -.  A  e.  Fin  /\  ( 1 ... ( N  +  1 ) )  e.  Fin )  ->  E. f  f : ( 1 ... ( N  +  1 ) ) -1-1-> A )
2926, 27, 28syl2anc 659 . . . . 5  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  E. f 
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A
)
30 erdsze2.a . . . . . . . 8  |-  ( ph  ->  A  C_  RR )
31 reex 9572 . . . . . . . 8  |-  RR  e.  _V
32 ssexg 4583 . . . . . . . 8  |-  ( ( A  C_  RR  /\  RR  e.  _V )  ->  A  e.  _V )
3330, 31, 32sylancl 660 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
3433adantr 463 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  A  e.  _V )
35 brdomg 7519 . . . . . 6  |-  ( A  e.  _V  ->  (
( 1 ... ( N  +  1 ) )  ~<_  A  <->  E. f 
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A
) )
3634, 35syl 16 . . . . 5  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
( 1 ... ( N  +  1 ) )  ~<_  A  <->  E. f 
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A
) )
3729, 36mpbird 232 . . . 4  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
1 ... ( N  + 
1 ) )  ~<_  A )
3825, 37pm2.61dan 789 . . 3  |-  ( ph  ->  ( 1 ... ( N  +  1 ) )  ~<_  A )
39 domeng 7523 . . . 4  |-  ( A  e.  _V  ->  (
( 1 ... ( N  +  1 ) )  ~<_  A  <->  E. s
( ( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) ) )
4033, 39syl 16 . . 3  |-  ( ph  ->  ( ( 1 ... ( N  +  1 ) )  ~<_  A  <->  E. s
( ( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) ) )
4138, 40mpbid 210 . 2  |-  ( ph  ->  E. s ( ( 1 ... ( N  +  1 ) ) 
~~  s  /\  s  C_  A ) )
42 simprr 755 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  C_  A )
4330adantr 463 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  A  C_  RR )
4442, 43sstrd 3499 . . . . 5  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  C_  RR )
45 ltso 9654 . . . . 5  |-  <  Or  RR
46 soss 4807 . . . . 5  |-  ( s 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  s ) )
4744, 45, 46mpisyl 18 . . . 4  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  <  Or  s )
48 fzfid 12065 . . . . 5  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
1 ... ( N  + 
1 ) )  e. 
Fin )
49 simprl 754 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
1 ... ( N  + 
1 ) )  ~~  s )
50 enfi 7729 . . . . . 6  |-  ( ( 1 ... ( N  +  1 ) ) 
~~  s  ->  (
( 1 ... ( N  +  1 ) )  e.  Fin  <->  s  e.  Fin ) )
5149, 50syl 16 . . . . 5  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
( 1 ... ( N  +  1 ) )  e.  Fin  <->  s  e.  Fin ) )
5248, 51mpbid 210 . . . 4  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  e.  Fin )
53 fz1iso 12495 . . . 4  |-  ( (  <  Or  s  /\  s  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )
5447, 52, 53syl2anc 659 . . 3  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  s ) )
55 isof1o 6196 . . . . . . . . . 10  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  s )  -> 
f : ( 1 ... ( # `  s
) ) -1-1-onto-> s )
5655adantl 464 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( # `  s
) ) -1-1-onto-> s )
57 hashen 12402 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... ( N  +  1 ) )  e.  Fin  /\  s  e.  Fin )  ->  ( ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( # `  s
)  <->  ( 1 ... ( N  +  1 ) )  ~~  s
) )
5848, 52, 57syl2anc 659 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
( # `  ( 1 ... ( N  + 
1 ) ) )  =  ( # `  s
)  <->  ( 1 ... ( N  +  1 ) )  ~~  s
) )
5949, 58mpbird 232 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 ( 1 ... ( N  +  1 ) ) )  =  ( # `  s
) )
6012adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
6159, 60eqtr3d 2497 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 s )  =  ( N  +  1 ) )
6261adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  ( # `
 s )  =  ( N  +  1 ) )
6362oveq2d 6286 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
1 ... ( # `  s
) )  =  ( 1 ... ( N  +  1 ) ) )
64 f1oeq2 5790 . . . . . . . . . 10  |-  ( ( 1 ... ( # `  s ) )  =  ( 1 ... ( N  +  1 ) )  ->  ( f : ( 1 ... ( # `  s
) ) -1-1-onto-> s  <->  f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> s ) )
6563, 64syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
f : ( 1 ... ( # `  s
) ) -1-1-onto-> s  <->  f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> s ) )
6656, 65mpbid 210 . . . . . . . 8  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> s )
67 f1of1 5797 . . . . . . . 8  |-  ( f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> s  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-> s )
6866, 67syl 16 . . . . . . 7  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-> s )
69 simplrr 760 . . . . . . 7  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  s  C_  A )
70 f1ss 5768 . . . . . . 7  |-  ( ( f : ( 1 ... ( N  + 
1 ) ) -1-1-> s  /\  s  C_  A
)  ->  f :
( 1 ... ( N  +  1 ) ) -1-1-> A )
7168, 69, 70syl2anc 659 . . . . . 6  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-> A )
72 simpr 459 . . . . . . . 8  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  s ) )
73 f1ofo 5805 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  s
) ) -1-1-onto-> s  ->  f : ( 1 ... ( # `
 s ) )
-onto-> s )
74 forn 5780 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  s
) ) -onto-> s  ->  ran  f  =  s
)
75 isoeq5 6194 . . . . . . . . 9  |-  ( ran  f  =  s  -> 
( f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  ran  f )  <->  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) ) )
7656, 73, 74, 754syl 21 . . . . . . . 8  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  ran  f )  <-> 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  s ) ) )
7772, 76mpbird 232 . . . . . . 7  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  ran  f ) )
78 isoeq4 6193 . . . . . . . 8  |-  ( ( 1 ... ( # `  s ) )  =  ( 1 ... ( N  +  1 ) )  ->  ( f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  ran  f )  <-> 
f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
7963, 78syl 16 . . . . . . 7  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  ran  f )  <-> 
f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
8077, 79mpbid 210 . . . . . 6  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( N  +  1 ) ) ,  ran  f
) )
8171, 80jca 530 . . . . 5  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f ) ) )
8281ex 432 . . . 4  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  s )  -> 
( f : ( 1 ... ( N  +  1 ) )
-1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  + 
1 ) ) ,  ran  f ) ) ) )
8382eximdv 1715 . . 3  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s )  ->  E. f
( f : ( 1 ... ( N  +  1 ) )
-1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  + 
1 ) ) ,  ran  f ) ) ) )
8454, 83mpd 15 . 2  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  E. f
( f : ( 1 ... ( N  +  1 ) )
-1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  + 
1 ) ) ,  ran  f ) ) )
8541, 84exlimddv 1731 1  |-  ( ph  ->  E. f ( f : ( 1 ... ( N  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823   _Vcvv 3106    C_ wss 3461   class class class wbr 4439    Or wor 4788   ran crn 4989   -1-1->wf1 5567   -onto->wfo 5568   -1-1-onto->wf1o 5569   ` cfv 5570    Isom wiso 5571  (class class class)co 6270    ~~ cen 7506    ~<_ cdom 7507   Fincfn 7509   RRcr 9480   1c1 9482    + caddc 9484    x. cmul 9486    < clt 9617    <_ cle 9618    - cmin 9796   NNcn 10531   NN0cn0 10791   ...cfz 11675   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388
This theorem is referenced by:  erdsze2  28913
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