MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  smobeth Structured version   Unicode version

Theorem smobeth 8961
Description: The beth function is strictly monotone. This function is not strictly the beth function, but rather bethA is the same as  ( card `  ( R1 `  ( om  +o  A ) ) ), since conventionally we start counting at the first infinite level, and ignore the finite levels. (Contributed by Mario Carneiro, 6-Jun-2013.) (Revised by Mario Carneiro, 2-Jun-2015.)
Assertion
Ref Expression
smobeth  |-  Smo  ( card  o.  R1 )

Proof of Theorem smobeth
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardf2 8324 . . . . . . 7  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 ffun 5733 . . . . . . 7  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  ->  Fun 
card )
31, 2ax-mp 5 . . . . . 6  |-  Fun  card
4 r1fnon 8185 . . . . . . 7  |-  R1  Fn  On
5 fnfun 5678 . . . . . . 7  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 5 . . . . . 6  |-  Fun  R1
7 funco 5626 . . . . . 6  |-  ( ( Fun  card  /\  Fun  R1 )  ->  Fun  ( card  o.  R1 ) )
83, 6, 7mp2an 672 . . . . 5  |-  Fun  ( card  o.  R1 )
9 funfn 5617 . . . . 5  |-  ( Fun  ( card  o.  R1 ) 
<->  ( card  o.  R1 )  Fn  dom  ( card 
o.  R1 ) )
108, 9mpbi 208 . . . 4  |-  ( card 
o.  R1 )  Fn 
dom  ( card  o.  R1 )
11 rnco 5513 . . . . 5  |-  ran  ( card  o.  R1 )  =  ran  ( card  |`  ran  R1 )
12 resss 5297 . . . . . . 7  |-  ( card  |` 
ran  R1 )  C_  card
13 rnss 5231 . . . . . . 7  |-  ( (
card  |`  ran  R1 ) 
C_  card  ->  ran  ( card  |` 
ran  R1 )  C_  ran  card )
1412, 13ax-mp 5 . . . . . 6  |-  ran  ( card 
|`  ran  R1 )  C_ 
ran  card
15 frn 5737 . . . . . . 7  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  ->  ran 
card  C_  On )
161, 15ax-mp 5 . . . . . 6  |-  ran  card  C_  On
1714, 16sstri 3513 . . . . 5  |-  ran  ( card 
|`  ran  R1 )  C_  On
1811, 17eqsstri 3534 . . . 4  |-  ran  ( card  o.  R1 )  C_  On
19 df-f 5592 . . . 4  |-  ( (
card  o.  R1 ) : dom  ( card  o.  R1 )
--> On  <->  ( ( card 
o.  R1 )  Fn 
dom  ( card  o.  R1 )  /\  ran  ( card 
o.  R1 )  C_  On ) )
2010, 18, 19mpbir2an 918 . . 3  |-  ( card 
o.  R1 ) : dom  ( card  o.  R1 )
--> On
21 dmco 5515 . . . 4  |-  dom  ( card  o.  R1 )  =  ( `' R1 " dom  card )
2221feq2i 5724 . . 3  |-  ( (
card  o.  R1 ) : dom  ( card  o.  R1 )
--> On  <->  ( card  o.  R1 ) : ( `' R1 " dom  card ) --> On )
2320, 22mpbi 208 . 2  |-  ( card 
o.  R1 ) : ( `' R1 " dom  card ) --> On
24 elpreima 6001 . . . . . . . . 9  |-  ( R1  Fn  On  ->  (
x  e.  ( `' R1 " dom  card ) 
<->  ( x  e.  On  /\  ( R1 `  x
)  e.  dom  card ) ) )
254, 24ax-mp 5 . . . . . . . 8  |-  ( x  e.  ( `' R1 " dom  card )  <->  ( x  e.  On  /\  ( R1
`  x )  e. 
dom  card ) )
2625simplbi 460 . . . . . . 7  |-  ( x  e.  ( `' R1 " dom  card )  ->  x  e.  On )
27 onelon 4903 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
2826, 27sylan 471 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  y  e.  On )
2925simprbi 464 . . . . . . . 8  |-  ( x  e.  ( `' R1 " dom  card )  ->  ( R1 `  x )  e. 
dom  card )
3029adantr 465 . . . . . . 7  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  x )  e.  dom  card )
31 r1ord2 8199 . . . . . . . . 9  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( R1 `  y
)  C_  ( R1 `  x ) ) )
3231imp 429 . . . . . . . 8  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  C_  ( R1 `  x ) )
3326, 32sylan 471 . . . . . . 7  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  C_  ( R1 `  x ) )
34 ssnum 8420 . . . . . . 7  |-  ( ( ( R1 `  x
)  e.  dom  card  /\  ( R1 `  y
)  C_  ( R1 `  x ) )  -> 
( R1 `  y
)  e.  dom  card )
3530, 33, 34syl2anc 661 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  e.  dom  card )
36 elpreima 6001 . . . . . . 7  |-  ( R1  Fn  On  ->  (
y  e.  ( `' R1 " dom  card ) 
<->  ( y  e.  On  /\  ( R1 `  y
)  e.  dom  card ) ) )
374, 36ax-mp 5 . . . . . 6  |-  ( y  e.  ( `' R1 " dom  card )  <->  ( y  e.  On  /\  ( R1
`  y )  e. 
dom  card ) )
3828, 35, 37sylanbrc 664 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  y  e.  ( `' R1 " dom  card ) )
3938rgen2 2889 . . . 4  |-  A. x  e.  ( `' R1 " dom  card ) A. y  e.  x  y  e.  ( `' R1 " dom  card )
40 dftr5 4543 . . . 4  |-  ( Tr  ( `' R1 " dom  card )  <->  A. x  e.  ( `' R1 " dom  card ) A. y  e.  x  y  e.  ( `' R1 " dom  card ) )
4139, 40mpbir 209 . . 3  |-  Tr  ( `' R1 " dom  card )
42 cnvimass 5357 . . . . 5  |-  ( `' R1 " dom  card )  C_  dom  R1
43 dffn2 5732 . . . . . . 7  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
444, 43mpbi 208 . . . . . 6  |-  R1 : On
--> _V
4544fdmi 5736 . . . . 5  |-  dom  R1  =  On
4642, 45sseqtri 3536 . . . 4  |-  ( `' R1 " dom  card )  C_  On
47 epweon 6603 . . . 4  |-  _E  We  On
48 wess 4866 . . . 4  |-  ( ( `' R1 " dom  card )  C_  On  ->  (  _E  We  On  ->  _E  We  ( `' R1 " dom  card ) ) )
4946, 47, 48mp2 9 . . 3  |-  _E  We  ( `' R1 " dom  card )
50 df-ord 4881 . . 3  |-  ( Ord  ( `' R1 " dom  card )  <->  ( Tr  ( `' R1 " dom  card )  /\  _E  We  ( `' R1 " dom  card ) ) )
5141, 49, 50mpbir2an 918 . 2  |-  Ord  ( `' R1 " dom  card )
52 r1sdom 8192 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  ~<  ( R1 `  x ) )
5326, 52sylan 471 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  ~<  ( R1 `  x ) )
54 cardsdom2 8369 . . . . . . 7  |-  ( ( ( R1 `  y
)  e.  dom  card  /\  ( R1 `  x
)  e.  dom  card )  ->  ( ( card `  ( R1 `  y
) )  e.  (
card `  ( R1 `  x ) )  <->  ( R1 `  y )  ~<  ( R1 `  x ) ) )
5535, 30, 54syl2anc 661 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card `  ( R1 `  y ) )  e.  ( card `  ( R1 `  x ) )  <-> 
( R1 `  y
)  ~<  ( R1 `  x ) ) )
5653, 55mpbird 232 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( card `  ( R1 `  y
) )  e.  (
card `  ( R1 `  x ) ) )
57 fvco2 5942 . . . . . 6  |-  ( ( R1  Fn  On  /\  y  e.  On )  ->  ( ( card  o.  R1 ) `  y )  =  ( card `  ( R1 `  y ) ) )
584, 28, 57sylancr 663 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  y )  =  (
card `  ( R1 `  y ) ) )
5926adantr 465 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  x  e.  On )
60 fvco2 5942 . . . . . 6  |-  ( ( R1  Fn  On  /\  x  e.  On )  ->  ( ( card  o.  R1 ) `  x )  =  ( card `  ( R1 `  x ) ) )
614, 59, 60sylancr 663 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  x )  =  (
card `  ( R1 `  x ) ) )
6256, 58, 613eltr4d 2570 . . . 4  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
)
6362ex 434 . . 3  |-  ( x  e.  ( `' R1 " dom  card )  ->  (
y  e.  x  -> 
( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
) )
6463adantl 466 . 2  |-  ( ( y  e.  ( `' R1 " dom  card )  /\  x  e.  ( `' R1 " dom  card ) )  ->  (
y  e.  x  -> 
( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
) )
6523, 51, 64, 21issmo 7019 1  |-  Smo  ( card  o.  R1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   _Vcvv 3113    C_ wss 3476   class class class wbr 4447   Tr wtr 4540    _E cep 4789    We wwe 4837   Ord word 4877   Oncon0 4878   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002    o. ccom 5003   Fun wfun 5582    Fn wfn 5583   -->wf 5584   ` cfv 5588   Smo wsmo 7016    ~~ cen 7513    ~< csdm 7515   R1cr1 8180   cardccrd 8316
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
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-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-om 6685  df-smo 7017  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-r1 8182  df-card 8320
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator