Theorem smobeth 8466

Description: The beth function is strictly monotone. This function is not strictly the beth function, but rather beth_A 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.

Step	Hyp	Ref	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 )
3	1, 2	ax-mp 5	. . . . . 6 |- Fun card
4		r1fnon 7696	. . . . . . 7 |- R1 Fn On
5		fnfun 5545	. . . . . . 7 |- ( R1 Fn On -> Fun R1 )
6	4, 5	ax-mp 5	. . . . . 6 |- Fun R1
7		funco 5494	. . . . . 6 |- ( ( Fun card /\ Fun R1 ) -> Fun ( card o. R1 ) )
8	3, 6, 7	mp2an 655	. . . . 5 |- Fun ( card o. R1 )
9		funfn 5485	. . . . 5 |- ( Fun ( card o. R1 ) <-> ( card o. R1 ) Fn dom ( card o. R1 ) )
10	8, 9	mpbi 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 )
14	12, 13	ax-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 )
16	1, 15	ax-mp 5	. . . . . 6 |- ran card C_ On
17	14, 16	sstri 3359	. . . . 5 |- ran ( card |` ran R1 ) C_ On
18	11, 17	eqsstri 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 ) )
20	10, 18, 19	mpbir2an 888	. . 3 |- ( card o. R1 ) : dom ( card o. R1 ) --> On
21		dmco 5381	. . . 4 |- dom ( card o. R1 ) = ( `' R1 " dom card )
22	21	feq2i 5589	. . 3 |- ( ( card o. R1 ) : dom ( card o. R1 ) --> On <-> ( card o. R1 ) : ( `' R1 " dom card ) --> On )
23	20, 22	mpbi 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 ) ) )
25	4, 24	ax-mp 5	. . . . . . . 8 |- ( x e. ( `' R1 " dom card ) <-> ( x e. On /\ ( R1 ` x ) e. dom card ) )
26	25	simplbi 448	. . . . . . 7 |- ( x e. ( `' R1 " dom card ) -> x e. On )
27		onelon 4609	. . . . . . 7 |- ( ( x e. On /\ y e. x ) -> y e. On )
28	26, 27	sylan 459	. . . . . 6 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> y e. On )
29	25	simprbi 452	. . . . . . . 8 |- ( x e. ( `' R1 " dom card ) -> ( R1 ` x ) e. dom card )
30	29	adantr 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 ) ) )
32	31	imp 420	. . . . . . . 8 |- ( ( x e. On /\ y e. x ) -> ( R1 ` y ) C_ ( R1 ` x ) )
33	26, 32	sylan 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 )
35	30, 33, 34	syl2anc 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 ) ) )
37	4, 36	ax-mp 5	. . . . . 6 |- ( y e. ( `' R1 " dom card ) <-> ( y e. On /\ ( R1 ` y ) e. dom card ) )
38	28, 35, 37	sylanbrc 647	. . . . 5 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> y e. ( `' R1 " dom card ) )
39	38	rgen2 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 ) )
41	39, 40	mpbir 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 )
44	4, 43	mpbi 201	. . . . . 6 |- R1 : On --> _V
45	44	fdmi 5599	. . . . 5 |- dom R1 = On
46	42, 45	sseqtri 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 ) ) )
49	46, 47, 48	mp2 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 ) ) )
51	41, 49, 50	mpbir2an 888	. 2 |- Ord ( `' R1 " dom card )
52		r1sdom 7703	. . . . . . 7 |- ( ( x e. On /\ y e. x ) -> ( R1 ` y ) ~< ( R1 ` x ) )
53	26, 52	sylan 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 ) ) )
55	35, 30, 54	syl2anc 644	. . . . . 6 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card ` ( R1 ` y ) ) e. ( card ` ( R1 ` x ) ) <-> ( R1 ` y ) ~< ( R1 ` x ) ) )
56	53, 55	mpbird 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 ) ) )
58	4, 28, 57	sylancr 646	. . . . 5 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card o. R1 ) ` y ) = ( card ` ( R1 ` y ) ) )
59	26	adantr 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 ) ) )
61	4, 59, 60	sylancr 646	. . . . 5 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card o. R1 ) ` x ) = ( card ` ( R1 ` x ) ) )
62	56, 58, 61	3eltr4d 2519	. . . 4 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card o. R1 ) ` y ) e. ( ( card o. R1 ) ` x ) )
63	62	ex 425	. . 3 |- ( x e. ( `' R1 " dom card ) -> ( y e. x -> ( ( card o. R1 ) ` y ) e. ( ( card o. R1 ) ` x ) ) )
64	63	adantl 454	. 2 |- ( ( y e. ( `' R1 " dom card ) /\ x e. ( `' R1 " dom card ) ) -> ( y e. x -> ( ( card o. R1 ) ` y ) e. ( ( card o. R1 ) ` x ) ) )
65	23, 51, 64, 21	issmo 6613	1 |- Smo ( card o. R1 )

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