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Theorem smobeth 8298
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 7666 . . . . . . 7  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 ffun 5474 . . . . . . 7  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  ->  Fun 
card )
31, 2ax-mp 8 . . . . . 6  |-  Fun  card
4 r1fnon 7529 . . . . . . 7  |-  R1  Fn  On
5 fnfun 5423 . . . . . . 7  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 8 . . . . . 6  |-  Fun  R1
7 funco 5374 . . . . . 6  |-  ( ( Fun  card  /\  Fun  R1 )  ->  Fun  ( card  o.  R1 ) )
83, 6, 7mp2an 653 . . . . 5  |-  Fun  ( card  o.  R1 )
9 funfn 5365 . . . . 5  |-  ( Fun  ( card  o.  R1 ) 
<->  ( card  o.  R1 )  Fn  dom  ( card 
o.  R1 ) )
108, 9mpbi 199 . . . 4  |-  ( card 
o.  R1 )  Fn 
dom  ( card  o.  R1 )
11 rnco 5261 . . . . 5  |-  ran  ( card  o.  R1 )  =  ran  ( card  |`  ran  R1 )
12 resss 5061 . . . . . . 7  |-  ( card  |` 
ran  R1 )  C_  card
13 rnss 4989 . . . . . . 7  |-  ( (
card  |`  ran  R1 ) 
C_  card  ->  ran  ( card  |` 
ran  R1 )  C_  ran  card )
1412, 13ax-mp 8 . . . . . 6  |-  ran  ( card 
|`  ran  R1 )  C_ 
ran  card
15 frn 5478 . . . . . . 7  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  ->  ran 
card  C_  On )
161, 15ax-mp 8 . . . . . 6  |-  ran  card  C_  On
1714, 16sstri 3264 . . . . 5  |-  ran  ( card 
|`  ran  R1 )  C_  On
1811, 17eqsstri 3284 . . . 4  |-  ran  ( card  o.  R1 )  C_  On
19 df-f 5341 . . . 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 886 . . 3  |-  ( card 
o.  R1 ) : dom  ( card  o.  R1 )
--> On
21 dmco 5263 . . . 4  |-  dom  ( card  o.  R1 )  =  ( `' R1 " dom  card )
2221feq2i 5467 . . 3  |-  ( (
card  o.  R1 ) : dom  ( card  o.  R1 )
--> On  <->  ( card  o.  R1 ) : ( `' R1 " dom  card ) --> On )
2320, 22mpbi 199 . 2  |-  ( card 
o.  R1 ) : ( `' R1 " dom  card ) --> On
24 elpreima 5728 . . . . . . . . 9  |-  ( R1  Fn  On  ->  (
x  e.  ( `' R1 " dom  card ) 
<->  ( x  e.  On  /\  ( R1 `  x
)  e.  dom  card ) ) )
254, 24ax-mp 8 . . . . . . . 8  |-  ( x  e.  ( `' R1 " dom  card )  <->  ( x  e.  On  /\  ( R1
`  x )  e. 
dom  card ) )
2625simplbi 446 . . . . . . 7  |-  ( x  e.  ( `' R1 " dom  card )  ->  x  e.  On )
27 onelon 4499 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
2826, 27sylan 457 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  y  e.  On )
2925simprbi 450 . . . . . . . 8  |-  ( x  e.  ( `' R1 " dom  card )  ->  ( R1 `  x )  e. 
dom  card )
3029adantr 451 . . . . . . 7  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  x )  e.  dom  card )
31 r1ord2 7543 . . . . . . . . 9  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( R1 `  y
)  C_  ( R1 `  x ) ) )
3231imp 418 . . . . . . . 8  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  C_  ( R1 `  x ) )
3326, 32sylan 457 . . . . . . 7  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  C_  ( R1 `  x ) )
34 ssnum 7756 . . . . . . 7  |-  ( ( ( R1 `  x
)  e.  dom  card  /\  ( R1 `  y
)  C_  ( R1 `  x ) )  -> 
( R1 `  y
)  e.  dom  card )
3530, 33, 34syl2anc 642 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  e.  dom  card )
36 elpreima 5728 . . . . . . 7  |-  ( R1  Fn  On  ->  (
y  e.  ( `' R1 " dom  card ) 
<->  ( y  e.  On  /\  ( R1 `  y
)  e.  dom  card ) ) )
374, 36ax-mp 8 . . . . . 6  |-  ( y  e.  ( `' R1 " dom  card )  <->  ( y  e.  On  /\  ( R1
`  y )  e. 
dom  card ) )
3828, 35, 37sylanbrc 645 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  y  e.  ( `' R1 " dom  card ) )
3938rgen2 2715 . . . 4  |-  A. x  e.  ( `' R1 " dom  card ) A. y  e.  x  y  e.  ( `' R1 " dom  card )
40 dftr5 4197 . . . 4  |-  ( Tr  ( `' R1 " dom  card )  <->  A. x  e.  ( `' R1 " dom  card ) A. y  e.  x  y  e.  ( `' R1 " dom  card ) )
4139, 40mpbir 200 . . 3  |-  Tr  ( `' R1 " dom  card )
42 cnvimass 5115 . . . . 5  |-  ( `' R1 " dom  card )  C_  dom  R1
43 dffn2 5473 . . . . . . 7  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
444, 43mpbi 199 . . . . . 6  |-  R1 : On
--> _V
4544fdmi 5477 . . . . 5  |-  dom  R1  =  On
4642, 45sseqtri 3286 . . . 4  |-  ( `' R1 " dom  card )  C_  On
47 epweon 4657 . . . 4  |-  _E  We  On
48 wess 4462 . . . 4  |-  ( ( `' R1 " dom  card )  C_  On  ->  (  _E  We  On  ->  _E  We  ( `' R1 " dom  card ) ) )
4946, 47, 48mp2 17 . . 3  |-  _E  We  ( `' R1 " dom  card )
50 df-ord 4477 . . 3  |-  ( Ord  ( `' R1 " dom  card )  <->  ( Tr  ( `' R1 " dom  card )  /\  _E  We  ( `' R1 " dom  card ) ) )
5141, 49, 50mpbir2an 886 . 2  |-  Ord  ( `' R1 " dom  card )
52 r1sdom 7536 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  ~<  ( R1 `  x ) )
5326, 52sylan 457 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  ~<  ( R1 `  x ) )
54 cardsdom2 7711 . . . . . . 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 642 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card `  ( R1 `  y ) )  e.  ( card `  ( R1 `  x ) )  <-> 
( R1 `  y
)  ~<  ( R1 `  x ) ) )
5653, 55mpbird 223 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( card `  ( R1 `  y
) )  e.  (
card `  ( R1 `  x ) ) )
57 fvco2 5677 . . . . . . 7  |-  ( ( R1  Fn  On  /\  y  e.  On )  ->  ( ( card  o.  R1 ) `  y )  =  ( card `  ( R1 `  y ) ) )
584, 28, 57sylancr 644 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  y )  =  (
card `  ( R1 `  y ) ) )
5926adantr 451 . . . . . . 7  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  x  e.  On )
60 fvco2 5677 . . . . . . 7  |-  ( ( R1  Fn  On  /\  x  e.  On )  ->  ( ( card  o.  R1 ) `  x )  =  ( card `  ( R1 `  x ) ) )
614, 59, 60sylancr 644 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  x )  =  (
card `  ( R1 `  x ) ) )
6258, 61eleq12d 2426 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( (
( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )  <->  (
card `  ( R1 `  y ) )  e.  ( card `  ( R1 `  x ) ) ) )
6356, 62mpbird 223 . . . 4  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
)
6463ex 423 . . 3  |-  ( x  e.  ( `' R1 " dom  card )  ->  (
y  e.  x  -> 
( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
) )
6564adantl 452 . 2  |-  ( ( y  e.  ( `' R1 " dom  card )  /\  x  e.  ( `' R1 " dom  card ) )  ->  (
y  e.  x  -> 
( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
) )
6623, 51, 65, 21issmo 6452 1  |-  Smo  ( card  o.  R1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619   E.wrex 2620   _Vcvv 2864    C_ wss 3228   class class class wbr 4104   Tr wtr 4194    _E cep 4385    We wwe 4433   Ord word 4473   Oncon0 4474   `'ccnv 4770   dom cdm 4771   ran crn 4772    |` cres 4773   "cima 4774    o. ccom 4775   Fun wfun 5331    Fn wfn 5332   -->wf 5333   ` cfv 5337   Smo wsmo 6449    ~~ cen 6948    ~< csdm 6950   R1cr1 7524   cardccrd 7658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-int 3944  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-se 4435  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-isom 5346  df-riota 6391  df-smo 6450  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-r1 7526  df-card 7662
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