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Theorem smobeth 8466
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 7835 . . . . . . 7  |-  card : {
x  |  E. y  e.  On  y  ~~  x }
--> On
2 ffun 5596 . . . . . . 7  |-  ( card
: { x  |  E. y  e.  On  y  ~~  x } --> On  ->  Fun 
card )
31, 2ax-mp 5 . . . . . 6  |-  Fun  card
4 r1fnon 7696 . . . . . . 7  |-  R1  Fn  On
5 fnfun 5545 . . . . . . 7  |-  ( R1  Fn  On  ->  Fun  R1 )
64, 5ax-mp 5 . . . . . 6  |-  Fun  R1
7 funco 5494 . . . . . 6  |-  ( ( Fun  card  /\  Fun  R1 )  ->  Fun  ( card  o.  R1 ) )
83, 6, 7mp2an 655 . . . . 5  |-  Fun  ( card  o.  R1 )
9 funfn 5485 . . . . 5  |-  ( Fun  ( card  o.  R1 ) 
<->  ( card  o.  R1 )  Fn  dom  ( card 
o.  R1 ) )
108, 9mpbi 201 . . . 4  |-  ( card 
o.  R1 )  Fn 
dom  ( card  o.  R1 )
11 rnco 5379 . . . . 5  |-  ran  ( card  o.  R1 )  =  ran  ( card  |`  ran  R1 )
12 resss 5173 . . . . . . 7  |-  ( card  |` 
ran  R1 )  C_  card
13 rnss 5101 . . . . . . 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 5600 . . . . . . 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 3359 . . . . 5  |-  ran  ( card 
|`  ran  R1 )  C_  On
1811, 17eqsstri 3380 . . . 4  |-  ran  ( card  o.  R1 )  C_  On
19 df-f 5461 . . . 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 888 . . 3  |-  ( card 
o.  R1 ) : dom  ( card  o.  R1 )
--> On
21 dmco 5381 . . . 4  |-  dom  ( card  o.  R1 )  =  ( `' R1 " dom  card )
2221feq2i 5589 . . 3  |-  ( (
card  o.  R1 ) : dom  ( card  o.  R1 )
--> On  <->  ( card  o.  R1 ) : ( `' R1 " dom  card ) --> On )
2320, 22mpbi 201 . 2  |-  ( card 
o.  R1 ) : ( `' R1 " dom  card ) --> On
24 elpreima 5853 . . . . . . . . 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 448 . . . . . . 7  |-  ( x  e.  ( `' R1 " dom  card )  ->  x  e.  On )
27 onelon 4609 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
2826, 27sylan 459 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  y  e.  On )
2925simprbi 452 . . . . . . . 8  |-  ( x  e.  ( `' R1 " dom  card )  ->  ( R1 `  x )  e. 
dom  card )
3029adantr 453 . . . . . . 7  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  x )  e.  dom  card )
31 r1ord2 7710 . . . . . . . . 9  |-  ( x  e.  On  ->  (
y  e.  x  -> 
( R1 `  y
)  C_  ( R1 `  x ) ) )
3231imp 420 . . . . . . . 8  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  C_  ( R1 `  x ) )
3326, 32sylan 459 . . . . . . 7  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  C_  ( R1 `  x ) )
34 ssnum 7925 . . . . . . 7  |-  ( ( ( R1 `  x
)  e.  dom  card  /\  ( R1 `  y
)  C_  ( R1 `  x ) )  -> 
( R1 `  y
)  e.  dom  card )
3530, 33, 34syl2anc 644 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  e.  dom  card )
36 elpreima 5853 . . . . . . 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 647 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  y  e.  ( `' R1 " dom  card ) )
3938rgen2 2804 . . . 4  |-  A. x  e.  ( `' R1 " dom  card ) A. y  e.  x  y  e.  ( `' R1 " dom  card )
40 dftr5 4308 . . . 4  |-  ( Tr  ( `' R1 " dom  card )  <->  A. x  e.  ( `' R1 " dom  card ) A. y  e.  x  y  e.  ( `' R1 " dom  card ) )
4139, 40mpbir 202 . . 3  |-  Tr  ( `' R1 " dom  card )
42 cnvimass 5227 . . . . 5  |-  ( `' R1 " dom  card )  C_  dom  R1
43 dffn2 5595 . . . . . . 7  |-  ( R1  Fn  On  <->  R1 : On
--> _V )
444, 43mpbi 201 . . . . . 6  |-  R1 : On
--> _V
4544fdmi 5599 . . . . 5  |-  dom  R1  =  On
4642, 45sseqtri 3382 . . . 4  |-  ( `' R1 " dom  card )  C_  On
47 epweon 4767 . . . 4  |-  _E  We  On
48 wess 4572 . . . 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 4587 . . 3  |-  ( Ord  ( `' R1 " dom  card )  <->  ( Tr  ( `' R1 " dom  card )  /\  _E  We  ( `' R1 " dom  card ) ) )
5141, 49, 50mpbir2an 888 . 2  |-  Ord  ( `' R1 " dom  card )
52 r1sdom 7703 . . . . . . 7  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( R1 `  y
)  ~<  ( R1 `  x ) )
5326, 52sylan 459 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( R1 `  y )  ~<  ( R1 `  x ) )
54 cardsdom2 7880 . . . . . . 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 644 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card `  ( R1 `  y ) )  e.  ( card `  ( R1 `  x ) )  <-> 
( R1 `  y
)  ~<  ( R1 `  x ) ) )
5653, 55mpbird 225 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( card `  ( R1 `  y
) )  e.  (
card `  ( R1 `  x ) ) )
57 fvco2 5801 . . . . . 6  |-  ( ( R1  Fn  On  /\  y  e.  On )  ->  ( ( card  o.  R1 ) `  y )  =  ( card `  ( R1 `  y ) ) )
584, 28, 57sylancr 646 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  y )  =  (
card `  ( R1 `  y ) ) )
5926adantr 453 . . . . . 6  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  x  e.  On )
60 fvco2 5801 . . . . . 6  |-  ( ( R1  Fn  On  /\  x  e.  On )  ->  ( ( card  o.  R1 ) `  x )  =  ( card `  ( R1 `  x ) ) )
614, 59, 60sylancr 646 . . . . 5  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  x )  =  (
card `  ( R1 `  x ) ) )
6256, 58, 613eltr4d 2519 . . . 4  |-  ( ( x  e.  ( `' R1 " dom  card )  /\  y  e.  x
)  ->  ( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
)
6362ex 425 . . 3  |-  ( x  e.  ( `' R1 " dom  card )  ->  (
y  e.  x  -> 
( ( card  o.  R1 ) `  y )  e.  ( ( card  o.  R1 ) `  x )
) )
6463adantl 454 . 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 6613 1  |-  Smo  ( card  o.  R1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322   class class class wbr 4215   Tr wtr 4305    _E cep 4495    We wwe 4543   Ord word 4583   Oncon0 4584   `'ccnv 4880   dom cdm 4881   ran crn 4882    |` cres 4883   "cima 4884    o. ccom 4885   Fun wfun 5451    Fn wfn 5452   -->wf 5453   ` cfv 5457   Smo wsmo 6610    ~~ cen 7109    ~< csdm 7111   R1cr1 7691   cardccrd 7827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-riota 6552  df-smo 6611  df-recs 6636  df-rdg 6671  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-r1 7693  df-card 7831
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