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Theorem erdsze2lem1 24879
Description: Lemma for erdsze2 24881. (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 10251 . . . . . . . . . . 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 10251 . . . . . . . . . . 11  |-  ( S  e.  NN  ->  ( S  -  1 )  e.  NN0 )
75, 6syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( S  -  1 )  e.  NN0 )
84, 7nn0mulcld 10269 . . . . . . . . 9  |-  ( ph  ->  ( ( R  - 
1 )  x.  ( S  -  1 ) )  e.  NN0 )
91, 8syl5eqel 2519 . . . . . . . 8  |-  ( ph  ->  N  e.  NN0 )
10 peano2nn0 10250 . . . . . . . 8  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
11 hashfz1 11620 . . . . . . . 8  |-  ( ( N  +  1 )  e.  NN0  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
129, 10, 113syl 19 . . . . . . 7  |-  ( ph  ->  ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( N  + 
1 ) )
1312adantr 452 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
14 erdsze2lem.l . . . . . . . 8  |-  ( ph  ->  N  <  ( # `  A ) )
1514adantr 452 . . . . . . 7  |-  ( (
ph  /\  A  e.  Fin )  ->  N  < 
( # `  A ) )
16 hashcl 11629 . . . . . . . 8  |-  ( A  e.  Fin  ->  ( # `
 A )  e. 
NN0 )
17 nn0ltp1le 10322 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  ( # `  A )  e.  NN0 )  -> 
( N  <  ( # `
 A )  <->  ( N  +  1 )  <_ 
( # `  A ) ) )
189, 16, 17syl2an 464 . . . . . . 7  |-  ( (
ph  /\  A  e.  Fin )  ->  ( N  <  ( # `  A
)  <->  ( N  + 
1 )  <_  ( # `
 A ) ) )
1915, 18mpbid 202 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( N  +  1 )  <_ 
( # `  A ) )
2013, 19eqbrtrd 4224 . . . . 5  |-  ( (
ph  /\  A  e.  Fin )  ->  ( # `  ( 1 ... ( N  +  1 ) ) )  <_  ( # `
 A ) )
21 fzfid 11302 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  ( 1 ... ( N  + 
1 ) )  e. 
Fin )
22 simpr 448 . . . . . 6  |-  ( (
ph  /\  A  e.  Fin )  ->  A  e. 
Fin )
23 hashdom 11643 . . . . . 6  |-  ( ( ( 1 ... ( N  +  1 ) )  e.  Fin  /\  A  e.  Fin )  ->  ( ( # `  (
1 ... ( N  + 
1 ) ) )  <_  ( # `  A
)  <->  ( 1 ... ( N  +  1 ) )  ~<_  A ) )
2421, 22, 23syl2anc 643 . . . . 5  |-  ( (
ph  /\  A  e.  Fin )  ->  ( (
# `  ( 1 ... ( N  +  1 ) ) )  <_ 
( # `  A )  <-> 
( 1 ... ( N  +  1 ) )  ~<_  A ) )
2520, 24mpbid 202 . . . 4  |-  ( (
ph  /\  A  e.  Fin )  ->  ( 1 ... ( N  + 
1 ) )  ~<_  A )
26 simpr 448 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  -.  A  e.  Fin )
27 fzfid 11302 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
1 ... ( N  + 
1 ) )  e. 
Fin )
28 isinffi 7869 . . . . . 6  |-  ( ( -.  A  e.  Fin  /\  ( 1 ... ( N  +  1 ) )  e.  Fin )  ->  E. f  f : ( 1 ... ( N  +  1 ) ) -1-1-> A )
2926, 27, 28syl2anc 643 . . . . 5  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  E. f 
f : ( 1 ... ( N  + 
1 ) ) -1-1-> A
)
30 erdsze2.a . . . . . . . 8  |-  ( ph  ->  A  C_  RR )
31 reex 9071 . . . . . . . 8  |-  RR  e.  _V
32 ssexg 4341 . . . . . . . 8  |-  ( ( A  C_  RR  /\  RR  e.  _V )  ->  A  e.  _V )
3330, 31, 32sylancl 644 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
3433adantr 452 . . . . . 6  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  A  e.  _V )
35 brdomg 7110 . . . . . 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 224 . . . 4  |-  ( (
ph  /\  -.  A  e.  Fin )  ->  (
1 ... ( N  + 
1 ) )  ~<_  A )
3825, 37pm2.61dan 767 . . 3  |-  ( ph  ->  ( 1 ... ( N  +  1 ) )  ~<_  A )
39 domeng 7114 . . . 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 202 . 2  |-  ( ph  ->  E. s ( ( 1 ... ( N  +  1 ) ) 
~~  s  /\  s  C_  A ) )
42 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  C_  A )
4330adantr 452 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  A  C_  RR )
4442, 43sstrd 3350 . . . . 5  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  C_  RR )
45 ltso 9146 . . . . 5  |-  <  Or  RR
46 soss 4513 . . . . 5  |-  ( s 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  s ) )
4744, 45, 46ee10 1385 . . . 4  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  <  Or  s )
48 fzfid 11302 . . . . 5  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
1 ... ( N  + 
1 ) )  e. 
Fin )
49 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
1 ... ( N  + 
1 ) )  ~~  s )
50 enfi 7317 . . . . . 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 202 . . . 4  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  s  e.  Fin )
53 fz1iso 11701 . . . 4  |-  ( (  <  Or  s  /\  s  e.  Fin )  ->  E. f  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )
5447, 52, 53syl2anc 643 . . 3  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  E. f 
f  Isom  <  ,  <  ( ( 1 ... ( # `
 s ) ) ,  s ) )
55 isof1o 6037 . . . . . . . . . 10  |-  ( f 
Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  s )  -> 
f : ( 1 ... ( # `  s
) ) -1-1-onto-> s )
5655adantl 453 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( # `  s
) ) -1-1-onto-> s )
57 hashen 11621 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... ( N  +  1 ) )  e.  Fin  /\  s  e.  Fin )  ->  ( ( # `  (
1 ... ( N  + 
1 ) ) )  =  ( # `  s
)  <->  ( 1 ... ( N  +  1 ) )  ~~  s
) )
5848, 52, 57syl2anc 643 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  (
( # `  ( 1 ... ( N  + 
1 ) ) )  =  ( # `  s
)  <->  ( 1 ... ( N  +  1 ) )  ~~  s
) )
5949, 58mpbird 224 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 ( 1 ... ( N  +  1 ) ) )  =  ( # `  s
) )
6012adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 ( 1 ... ( N  +  1 ) ) )  =  ( N  +  1 ) )
6159, 60eqtr3d 2469 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( (
1 ... ( N  + 
1 ) )  ~~  s  /\  s  C_  A
) )  ->  ( # `
 s )  =  ( N  +  1 ) )
6261adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  ( # `
 s )  =  ( N  +  1 ) )
6362oveq2d 6089 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  (
1 ... ( # `  s
) )  =  ( 1 ... ( N  +  1 ) ) )
64 f1oeq2 5658 . . . . . . . . . 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 202 . . . . . . . 8  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-onto-> s )
67 f1of1 5665 . . . . . . . 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 738 . . . . . . 7  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  s  C_  A )
70 f1ss 5636 . . . . . . 7  |-  ( ( f : ( 1 ... ( N  + 
1 ) ) -1-1-> s  /\  s  C_  A
)  ->  f :
( 1 ... ( N  +  1 ) ) -1-1-> A )
7168, 69, 70syl2anc 643 . . . . . 6  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f : ( 1 ... ( N  +  1 ) ) -1-1-> A )
72 simpr 448 . . . . . . . 8  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  s ) )
73 f1ofo 5673 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  s
) ) -1-1-onto-> s  ->  f : ( 1 ... ( # `
 s ) )
-onto-> s )
74 forn 5648 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  s
) ) -onto-> s  ->  ran  f  =  s
)
7556, 73, 743syl 19 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  ran  f  =  s )
76 isoeq5 6035 . . . . . . . . 9  |-  ( ran  f  =  s  -> 
( f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  ran  f )  <->  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) ) )
7775, 76syl 16 . . . . . . . 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 ) ) )
7872, 77mpbird 224 . . . . . . 7  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  ran  f ) )
79 isoeq4 6034 . . . . . . . 8  |-  ( ( 1 ... ( # `  s ) )  =  ( 1 ... ( N  +  1 ) )  ->  ( f  Isom  <  ,  <  (
( 1 ... ( # `
 s ) ) ,  ran  f )  <-> 
f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
8063, 79syl 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
) ) )
8178, 80mpbid 202 . . . . . 6  |-  ( ( ( ph  /\  (
( 1 ... ( N  +  1 ) )  ~~  s  /\  s  C_  A ) )  /\  f  Isom  <  ,  <  ( ( 1 ... ( # `  s
) ) ,  s ) )  ->  f  Isom  <  ,  <  (
( 1 ... ( N  +  1 ) ) ,  ran  f
) )
8271, 81jca 519 . . . . 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 ) ) )
8382ex 424 . . . 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 ) ) ) )
8483eximdv 1632 . . 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 ) ) ) )
8554, 84mpd 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 ) ) )
8641, 85exlimddv 1648 1  |-  ( ph  ->  E. f ( f : ( 1 ... ( N  +  1 ) ) -1-1-> A  /\  f  Isom  <  ,  <  ( ( 1 ... ( N  +  1 ) ) ,  ran  f
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948    C_ wss 3312   class class class wbr 4204    Or wor 4494   ran crn 4871   -1-1->wf1 5443   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446    Isom wiso 5447  (class class class)co 6073    ~~ cen 7098    ~<_ cdom 7099   Fincfn 7101   RRcr 8979   1c1 8981    + caddc 8983    x. cmul 8985    < clt 9110    <_ cle 9111    - cmin 9281   NNcn 9990   NN0cn0 10211   ...cfz 11033   #chash 11608
This theorem is referenced by:  erdsze2  24881
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  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-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7469  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-hash 11609
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