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Theorem smobeth 8755
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 8118 . . . . . . 7  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 ffun 5566 . . . . . . 7  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  ->  Fun 
card )
31, 2ax-mp 5 . . . . . 6  |-  Fun  card
4 r1fnon 7979 . . . . . . 7  |-  R1  Fn  On
5 fnfun 5513 . . . . . . 7  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 5 . . . . . 6  |-  Fun  R1
7 funco 5461 . . . . . 6  |-  ( ( Fun  card  /\  Fun  R1 )  ->  Fun  ( card  o.  R1 ) )
83, 6, 7mp2an 672 . . . . 5  |-  Fun  ( card  o.  R1 )
9 funfn 5452 . . . . 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 5349 . . . . 5  |-  ran  ( card  o.  R1 )  =  ran  ( card  |`  ran  R1 )
12 resss 5139 . . . . . . 7  |-  ( card  |` 
ran  R1 )  C_  card
13 rnss 5073 . . . . . . 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 5570 . . . . . . 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 3370 . . . . 5  |-  ran  ( card 
|`  ran  R1 )  C_  On
1811, 17eqsstri 3391 . . . 4  |-  ran  ( card  o.  R1 )  C_  On
19 df-f 5427 . . . 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 911 . . 3  |-  ( card 
o.  R1 ) : dom  ( card  o.  R1 )
--> On
21 dmco 5351 . . . 4  |-  dom  ( card  o.  R1 )  =  ( `' R1 " dom  card )
2221feq2i 5557 . . 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 5828 . . . . . . . . 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 4749 . . . . . . 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 7993 . . . . . . . . 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 8214 . . . . . . 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 5828 . . . . . . 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 2817 . . . 4  |-  A. x  e.  ( `' R1 " dom  card ) A. y  e.  x  y  e.  ( `' R1 " dom  card )
40 dftr5 4393 . . . 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 5194 . . . . 5  |-  ( `' R1 " dom  card )  C_  dom  R1
43 dffn2 5565 . . . . . . 7  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
444, 43mpbi 208 . . . . . 6  |-  R1 : On
--> _V
4544fdmi 5569 . . . . 5  |-  dom  R1  =  On
4642, 45sseqtri 3393 . . . 4  |-  ( `' R1 " dom  card )  C_  On
47 epweon 6400 . . . 4  |-  _E  We  On
48 wess 4712 . . . 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 4727 . . 3  |-  ( Ord  ( `' R1 " dom  card )  <->  ( Tr  ( `' R1 " dom  card )  /\  _E  We  ( `' R1 " dom  card ) ) )
5141, 49, 50mpbir2an 911 . 2  |-  Ord  ( `' R1 " dom  card )
52 r1sdom 7986 . . . . . . 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 8163 . . . . . . 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 5771 . . . . . 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 5771 . . . . . 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 2524 . . . 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 6814 1  |-  Smo  ( card  o.  R1 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721   _Vcvv 2977    C_ wss 3333   class class class wbr 4297   Tr wtr 4390    _E cep 4635    We wwe 4683   Ord word 4723   Oncon0 4724   `'ccnv 4844   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848    o. ccom 4849   Fun wfun 5417    Fn wfn 5418   -->wf 5419   ` cfv 5423   Smo wsmo 6811    ~~ cen 7312    ~< csdm 7314   R1cr1 7974   cardccrd 8110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-om 6482  df-smo 6812  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-r1 7976  df-card 8114
This theorem is referenced by: (None)
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