Theorem smobeth 9408

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 8769	. . . . . . 7 |- card : { x | E. y e. On y ~~ x } --> On
2		ffun 6048	. . . . . . 7 |- ( card : { x | E. y e. On y ~~ x } --> On -> Fun card )
3	1, 2	ax-mp 5	. . . . . 6 |- Fun card
4		r1fnon 8630	. . . . . . 7 |- R1 Fn On
5		fnfun 5988	. . . . . . 7 |- ( R1 Fn On -> Fun R1 )
6	4, 5	ax-mp 5	. . . . . 6 |- Fun R1
7		funco 5928	. . . . . 6 |- ( ( Fun card /\ Fun R1 ) -> Fun ( card o. R1 ) )
8	3, 6, 7	mp2an 708	. . . . 5 |- Fun ( card o. R1 )
9		funfn 5918	. . . . 5 |- ( Fun ( card o. R1 ) <-> ( card o. R1 ) Fn dom ( card o. R1 ) )
10	8, 9	mpbi 220	. . . 4 |- ( card o. R1 ) Fn dom ( card o. R1 )
11		rnco 5641	. . . . 5 |- ran ( card o. R1 ) = ran ( card |` ran R1 )
12		resss 5422	. . . . . . 7 |- ( card |` ran R1 ) C_ card
13		rnss 5354	. . . . . . 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 6053	. . . . . . 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 3612	. . . . 5 |- ran ( card |` ran R1 ) C_ On
18	11, 17	eqsstri 3635	. . . 4 |- ran ( card o. R1 ) C_ On
19		df-f 5892	. . . 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 955	. . 3 |- ( card o. R1 ) : dom ( card o. R1 ) --> On
21		dmco 5643	. . . 4 |- dom ( card o. R1 ) = ( `' R1 " dom card )
22	21	feq2i 6037	. . 3 |- ( ( card o. R1 ) : dom ( card o. R1 ) --> On <-> ( card o. R1 ) : ( `' R1 " dom card ) --> On )
23	20, 22	mpbi 220	. 2 |- ( card o. R1 ) : ( `' R1 " dom card ) --> On
24		elpreima 6337	. . . . . . . . 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 476	. . . . . . 7 |- ( x e. ( `' R1 " dom card ) -> x e. On )
27		onelon 5748	. . . . . . 7 |- ( ( x e. On /\ y e. x ) -> y e. On )
28	26, 27	sylan 488	. . . . . 6 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> y e. On )
29	25	simprbi 480	. . . . . . . 8 |- ( x e. ( `' R1 " dom card ) -> ( R1 ` x ) e. dom card )
30	29	adantr 481	. . . . . . 7 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( R1 ` x ) e. dom card )
31		r1ord2 8644	. . . . . . . . 9 |- ( x e. On -> ( y e. x -> ( R1 ` y ) C_ ( R1 ` x ) ) )
32	31	imp 445	. . . . . . . 8 |- ( ( x e. On /\ y e. x ) -> ( R1 ` y ) C_ ( R1 ` x ) )
33	26, 32	sylan 488	. . . . . . 7 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( R1 ` y ) C_ ( R1 ` x ) )
34		ssnum 8862	. . . . . . 7 |- ( ( ( R1 ` x ) e. dom card /\ ( R1 ` y ) C_ ( R1 ` x ) ) -> ( R1 ` y ) e. dom card )
35	30, 33, 34	syl2anc 693	. . . . . 6 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( R1 ` y ) e. dom card )
36		elpreima 6337	. . . . . . 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 698	. . . . 5 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> y e. ( `' R1 " dom card ) )
39	38	rgen2 2975	. . . 4 |- A. x e. ( `' R1 " dom card ) A. y e. x y e. ( `' R1 " dom card )
40		dftr5 4755	. . . 4 |- ( Tr ( `' R1 " dom card ) <-> A. x e. ( `' R1 " dom card ) A. y e. x y e. ( `' R1 " dom card ) )
41	39, 40	mpbir 221	. . 3 |- Tr ( `' R1 " dom card )
42		cnvimass 5485	. . . . 5 |- ( `' R1 " dom card ) C_ dom R1
43		dffn2 6047	. . . . . . 7 |- ( R1 Fn On <-> R1 : On --> _V )
44	4, 43	mpbi 220	. . . . . 6 |- R1 : On --> _V
45	44	fdmi 6052	. . . . 5 |- dom R1 = On
46	42, 45	sseqtri 3637	. . . 4 |- ( `' R1 " dom card ) C_ On
47		epweon 6983	. . . 4 |- _E We On
48		wess 5101	. . . 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 5726	. . 3 |- ( Ord ( `' R1 " dom card ) <-> ( Tr ( `' R1 " dom card ) /\ _E We ( `' R1 " dom card ) ) )
51	41, 49, 50	mpbir2an 955	. 2 |- Ord ( `' R1 " dom card )
52		r1sdom 8637	. . . . . . 7 |- ( ( x e. On /\ y e. x ) -> ( R1 ` y ) ~< ( R1 ` x ) )
53	26, 52	sylan 488	. . . . . 6 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( R1 ` y ) ~< ( R1 ` x ) )
54		cardsdom2 8814	. . . . . . 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 693	. . . . . 6 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card ` ( R1 ` y ) ) e. ( card ` ( R1 ` x ) ) <-> ( R1 ` y ) ~< ( R1 ` x ) ) )
56	53, 55	mpbird 247	. . . . 5 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( card ` ( R1 ` y ) ) e. ( card ` ( R1 ` x ) ) )
57		fvco2 6273	. . . . . 6 |- ( ( R1 Fn On /\ y e. On ) -> ( ( card o. R1 ) ` y ) = ( card ` ( R1 ` y ) ) )
58	4, 28, 57	sylancr 695	. . . . 5 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card o. R1 ) ` y ) = ( card ` ( R1 ` y ) ) )
59	26	adantr 481	. . . . . 6 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> x e. On )
60		fvco2 6273	. . . . . 6 |- ( ( R1 Fn On /\ x e. On ) -> ( ( card o. R1 ) ` x ) = ( card ` ( R1 ` x ) ) )
61	4, 59, 60	sylancr 695	. . . . 5 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card o. R1 ) ` x ) = ( card ` ( R1 ` x ) ) )
62	56, 58, 61	3eltr4d 2716	. . . 4 |- ( ( x e. ( `' R1 " dom card ) /\ y e. x ) -> ( ( card o. R1 ) ` y ) e. ( ( card o. R1 ) ` x ) )
63	62	ex 450	. . 3 |- ( x e. ( `' R1 " dom card ) -> ( y e. x -> ( ( card o. R1 ) ` y ) e. ( ( card o. R1 ) ` x ) ) )
64	63	adantl 482	. 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 7445	1 |- Smo ( card o. R1 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   Tr wtr 4752   _E cep 5028   We wwe 5072   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Ord word 5722   Oncon0 5723   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   Smo wsmo 7442   ~~ cen 7952   ~< csdm 7954   R1cr1 8625   cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-smo 7443  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-r1 8627  df-card 8765

This theorem is referenced by: (None)