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Theorem smobeth 8421
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 7790 . . . . . . 7  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 ffun 5556 . . . . . . 7  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  ->  Fun 
card )
31, 2ax-mp 8 . . . . . 6  |-  Fun  card
4 r1fnon 7653 . . . . . . 7  |-  R1  Fn  On
5 fnfun 5505 . . . . . . 7  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 8 . . . . . 6  |-  Fun  R1
7 funco 5454 . . . . . 6  |-  ( ( Fun  card  /\  Fun  R1 )  ->  Fun  ( card  o.  R1 ) )
83, 6, 7mp2an 654 . . . . 5  |-  Fun  ( card  o.  R1 )
9 funfn 5445 . . . . 5  |-  ( Fun  ( card  o.  R1 ) 
<->  ( card  o.  R1 )  Fn  dom  ( card 
o.  R1 ) )
108, 9mpbi 200 . . . 4  |-  ( card 
o.  R1 )  Fn 
dom  ( card  o.  R1 )
11 rnco 5339 . . . . 5  |-  ran  ( card  o.  R1 )  =  ran  ( card  |`  ran  R1 )
12 resss 5133 . . . . . . 7  |-  ( card  |` 
ran  R1 )  C_  card
13 rnss 5061 . . . . . . 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 5560 . . . . . . 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 3321 . . . . 5  |-  ran  ( card 
|`  ran  R1 )  C_  On
1811, 17eqsstri 3342 . . . 4  |-  ran  ( card  o.  R1 )  C_  On
19 df-f 5421 . . . 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 887 . . 3  |-  ( card 
o.  R1 ) : dom  ( card  o.  R1 )
--> On
21 dmco 5341 . . . 4  |-  dom  ( card  o.  R1 )  =  ( `' R1 " dom  card )
2221feq2i 5549 . . 3  |-  ( (
card  o.  R1 ) : dom  ( card  o.  R1 )
--> On  <->  ( card  o.  R1 ) : ( `' R1 " dom  card ) --> On )
2320, 22mpbi 200 . 2  |-  ( card 
o.  R1 ) : ( `' R1 " dom  card ) --> On
24 elpreima 5813 . . . . . . . . 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 447 . . . . . . 7  |-  ( x  e.  ( `' R1 " dom  card )  ->  x  e.  On )
27 onelon 4570 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
2826, 27sylan 458 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  y  e.  On )
2925simprbi 451 . . . . . . . 8  |-  ( x  e.  ( `' R1 " dom  card )  ->  ( R1 `  x )  e. 
dom  card )
3029adantr 452 . . . . . . 7  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  x )  e.  dom  card )
31 r1ord2 7667 . . . . . . . . 9  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( R1 `  y
)  C_  ( R1 `  x ) ) )
3231imp 419 . . . . . . . 8  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  C_  ( R1 `  x ) )
3326, 32sylan 458 . . . . . . 7  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  C_  ( R1 `  x ) )
34 ssnum 7880 . . . . . . 7  |-  ( ( ( R1 `  x
)  e.  dom  card  /\  ( R1 `  y
)  C_  ( R1 `  x ) )  -> 
( R1 `  y
)  e.  dom  card )
3530, 33, 34syl2anc 643 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  e.  dom  card )
36 elpreima 5813 . . . . . . 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 646 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  y  e.  ( `' R1 " dom  card ) )
3938rgen2 2766 . . . 4  |-  A. x  e.  ( `' R1 " dom  card ) A. y  e.  x  y  e.  ( `' R1 " dom  card )
40 dftr5 4269 . . . 4  |-  ( Tr  ( `' R1 " dom  card )  <->  A. x  e.  ( `' R1 " dom  card ) A. y  e.  x  y  e.  ( `' R1 " dom  card ) )
4139, 40mpbir 201 . . 3  |-  Tr  ( `' R1 " dom  card )
42 cnvimass 5187 . . . . 5  |-  ( `' R1 " dom  card )  C_  dom  R1
43 dffn2 5555 . . . . . . 7  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
444, 43mpbi 200 . . . . . 6  |-  R1 : On
--> _V
4544fdmi 5559 . . . . 5  |-  dom  R1  =  On
4642, 45sseqtri 3344 . . . 4  |-  ( `' R1 " dom  card )  C_  On
47 epweon 4727 . . . 4  |-  _E  We  On
48 wess 4533 . . . 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 4548 . . 3  |-  ( Ord  ( `' R1 " dom  card )  <->  ( Tr  ( `' R1 " dom  card )  /\  _E  We  ( `' R1 " dom  card ) ) )
5141, 49, 50mpbir2an 887 . 2  |-  Ord  ( `' R1 " dom  card )
52 r1sdom 7660 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  ~<  ( R1 `  x ) )
5326, 52sylan 458 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  ~<  ( R1 `  x ) )
54 cardsdom2 7835 . . . . . . 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 643 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card `  ( R1 `  y ) )  e.  ( card `  ( R1 `  x ) )  <-> 
( R1 `  y
)  ~<  ( R1 `  x ) ) )
5653, 55mpbird 224 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( card `  ( R1 `  y
) )  e.  (
card `  ( R1 `  x ) ) )
57 fvco2 5761 . . . . . 6  |-  ( ( R1  Fn  On  /\  y  e.  On )  ->  ( ( card  o.  R1 ) `  y )  =  ( card `  ( R1 `  y ) ) )
584, 28, 57sylancr 645 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  y )  =  (
card `  ( R1 `  y ) ) )
5926adantr 452 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  x  e.  On )
60 fvco2 5761 . . . . . 6  |-  ( ( R1  Fn  On  /\  x  e.  On )  ->  ( ( card  o.  R1 ) `  x )  =  ( card `  ( R1 `  x ) ) )
614, 59, 60sylancr 645 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  x )  =  (
card `  ( R1 `  x ) ) )
6256, 58, 613eltr4d 2489 . . . 4  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
)
6362ex 424 . . 3  |-  ( x  e.  ( `' R1 " dom  card )  ->  (
y  e.  x  -> 
( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
) )
6463adantl 453 . 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 6573 1  |-  Smo  ( card  o.  R1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2394   A.wral 2670   E.wrex 2671   _Vcvv 2920    C_ wss 3284   class class class wbr 4176   Tr wtr 4266    _E cep 4456    We wwe 4504   Ord word 4544   Oncon0 4545   `'ccnv 4840   dom cdm 4841   ran crn 4842    |` cres 4843   "cima 4844    o. ccom 4845   Fun wfun 5411    Fn wfn 5412   -->wf 5413   ` cfv 5417   Smo wsmo 6570    ~~ cen 7069    ~< csdm 7071   R1cr1 7648   cardccrd 7782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6512  df-smo 6571  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-r1 7650  df-card 7786
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