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Theorem erdsze2lem1 28315
Description: Lemma for erdsze2 28317. (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 10837 . . . . . . . . . . 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 10837 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
75, 6syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( S  -  1 )  e.  NN0 )
84, 7nn0mulcld 10857 . . . . . . . . 9  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  e.  NN0 )
91, 8syl5eqel 2559 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
10 peano2nn0 10836 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
11 hashfz1 12387 . . . . . . . 8  |-  ( ( N  +  1 )  e.  NN0  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
129, 10, 113syl 20 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( N  + 
1 ) )
1312adantr 465 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
14 erdsze2lem.l . . . . . . . 8  |-  ( ph  ->  N  <  ( # `  A ) )
1514adantr 465 . . . . . . 7  |-  ( (
ph  /\  A  e.  Fin )  ->  N  < 
( # `  A ) )
16 hashcl 12396 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
17 nn0ltp1le 10920 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( # `  A )  e.  NN0 )  -> 
( N  <  ( # `
 A )  <->  ( N  +  1 )  <_ 
( # `  A ) ) )
189, 16, 17syl2an 477 . . . . . . 7  |-  ( (
ph  /\  A  e.  Fin )  ->  ( N  <  ( # `  A
)  <->  ( N  + 
1 )  <_  ( # `
 A ) ) )
1915, 18mpbid 210 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( N  +  1 )  <_ 
( # `  A ) )
2013, 19eqbrtrd 4467 . . . . 5  |-  ( (
ph  /\  A  e.  Fin )  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  <_  ( # `
 A ) )
21 fzfid 12051 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( 1 ... ( N  + 
1 ) )  e. 
Fin )
22 simpr 461 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  A  e. 
Fin )
23 hashdom 12415 . . . . . 6  |-  ( ( ( 1 ... ( N  +  1 ) )  e.  Fin  /\  A  e.  Fin )  ->  ( ( # `  (
1 ... ( N  + 
1 ) ) )  <_  ( # `  A
)  <->  ( 1 ... ( N  +  1 ) )  ~<_  A ) )
2421, 22, 23syl2anc 661 . . . . 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 461 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  -.  A  e.  Fin )
27 fzfid 12051 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
1 ... ( N  + 
1 ) )  e. 
Fin )
28 isinffi 8373 . . . . . 6  |-  ( ( -.  A  e.  Fin  /\  ( 1 ... ( N  +  1 ) )  e.  Fin )  ->  E. f  f : ( 1 ... ( N  +  1 ) ) -1-1-> A )
2926, 27, 28syl2anc 661 . . . . 5  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  E. f 
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A
)
30 erdsze2.a . . . . . . . 8  |-  ( ph  ->  A  C_  RR )
31 reex 9583 . . . . . . . 8  |-  RR  e.  _V
32 ssexg 4593 . . . . . . . 8  |-  ( ( A  C_  RR  /\  RR  e.  _V )  ->  A  e.  _V )
3330, 31, 32sylancl 662 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
3433adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  A  e.  _V )
35 brdomg 7526 . . . . . 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 7530 . . . 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 756 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  C_  A )
4330adantr 465 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  A  C_  RR )
4442, 43sstrd 3514 . . . . 5  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  C_  RR )
45 ltso 9665 . . . . 5  |-  <  Or  RR
46 soss 4818 . . . . 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 12051 . . . . 5  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
1 ... ( N  + 
1 ) )  e. 
Fin )
49 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
1 ... ( N  + 
1 ) )  ~~  s )
50 enfi 7736 . . . . . 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 12477 . . . 4  |-  ( (  <  Or  s  /\  s  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )
5447, 52, 53syl2anc 661 . . 3  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  s ) )
55 isof1o 6209 . . . . . . . . . 10  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  s )  -> 
f : ( 1 ... ( # `  s
) ) -1-1-onto-> s )
5655adantl 466 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( # `  s
) ) -1-1-onto-> s )
57 hashen 12388 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... ( N  +  1 ) )  e.  Fin  /\  s  e.  Fin )  ->  ( ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( # `  s
)  <->  ( 1 ... ( N  +  1 ) )  ~~  s
) )
5848, 52, 57syl2anc 661 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
6159, 60eqtr3d 2510 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 s )  =  ( N  +  1 ) )
6261adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  ( # `
 s )  =  ( N  +  1 ) )
6362oveq2d 6300 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
1 ... ( # `  s
) )  =  ( 1 ... ( N  +  1 ) ) )
64 f1oeq2 5808 . . . . . . . . . 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 5815 . . . . . . . 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 5786 . . . . . . 7  |-  ( ( f : ( 1 ... ( N  + 
1 ) ) -1-1-> s  /\  s  C_  A
)  ->  f :
( 1 ... ( N  +  1 ) ) -1-1-> A )
7168, 69, 70syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-> A )
72 simpr 461 . . . . . . . 8  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  s ) )
73 f1ofo 5823 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  s
) ) -1-1-onto-> s  ->  f : ( 1 ... ( # `
 s ) )
-onto-> s )
74 forn 5798 . . . . . . . . 9  |-  ( f : ( 1 ... ( # `  s
) ) -onto-> s  ->  ran  f  =  s
)
75 isoeq5 6207 . . . . . . . . 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 6206 . . . . . . . 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 532 . . . . 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 434 . . . 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 1686 . . 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 1702 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 369    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113    C_ wss 3476   class class class wbr 4447    Or wor 4799   ran crn 5000   -1-1->wf1 5585   -onto->wfo 5586   -1-1-onto->wf1o 5587   ` cfv 5588    Isom wiso 5589  (class class class)co 6284    ~~ cen 7513    ~<_ cdom 7514   Fincfn 7516   RRcr 9491   1c1 9493    + caddc 9495    x. cmul 9497    < clt 9628    <_ cle 9629    - cmin 9805   NNcn 10536   NN0cn0 10795   ...cfz 11672   #chash 12373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-hash 12374
This theorem is referenced by:  erdsze2  28317
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