Theorem cantnflem1b 8209

Description: Lemma for cantnf 8216. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)

Hypotheses

Ref	Expression
cantnfs.s	|- S = dom ( A CNF B )
cantnfs.a	|- ( ph -> A e. On )
cantnfs.b	|- ( ph -> B e. On )
oemapval.t	|- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) }
oemapval.f	|- ( ph -> F e. S )
oemapval.g	|- ( ph -> G e. S )
oemapvali.r	|- ( ph -> F T G )
oemapvali.x	|- X = U. { c e. B | ( F ` c ) e. ( G ` c ) }
cantnflem1.o	|- O = OrdIso ( _E , ( G supp (/) ) )

Assertion

Ref	Expression
cantnflem1b	|- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X C_ ( O ` u ) )

Distinct variable groups:   u, c, w, x, y, z, B   A, c, u, w, x, y, z   T, c, u   u, F, w, x, y, z   S, c, u, x, y, z   G, c, u, w, x, y, z   u, O, w, x, y, z   ph, u, x, y, z   u, X, w, x, y, z   F, c   ph, c

Allowed substitution hints:   ph( w)   S( w)   T( x, y, z, w)   O( c)   X( c)

Proof of Theorem cantnflem1b

Step	Hyp	Ref	Expression
1		simprr 774	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( `' O ` X ) C_ u )
2		cantnflem1.o	. . . . . . . 8 |- O = OrdIso ( _E , ( G supp (/) ) )
3	2	oicl 8062	. . . . . . 7 |- Ord dom O
4		cantnfs.b	. . . . . . . . . . . 12 |- ( ph -> B e. On )
5		suppssdm 6946	. . . . . . . . . . . . 13 |- ( G supp (/) ) C_ dom G
6		oemapval.g	. . . . . . . . . . . . . . . 16 |- ( ph -> G e. S )
7		cantnfs.s	. . . . . . . . . . . . . . . . 17 |- S = dom ( A CNF B )
8		cantnfs.a	. . . . . . . . . . . . . . . . 17 |- ( ph -> A e. On )
9	7, 8, 4	cantnfs 8189	. . . . . . . . . . . . . . . 16 |- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) )
10	6, 9	mpbid 215	. . . . . . . . . . . . . . 15 |- ( ph -> ( G : B --> A /\ G finSupp (/) ) )
11	10	simpld 466	. . . . . . . . . . . . . 14 |- ( ph -> G : B --> A )
12		fdm 5745	. . . . . . . . . . . . . 14 |- ( G : B --> A -> dom G = B )
13	11, 12	syl 17	. . . . . . . . . . . . 13 |- ( ph -> dom G = B )
14	5, 13	syl5sseq 3466	. . . . . . . . . . . 12 |- ( ph -> ( G supp (/) ) C_ B )
15	4, 14	ssexd 4543	. . . . . . . . . . 11 |- ( ph -> ( G supp (/) ) e. _V )
16	7, 8, 4, 2, 6	cantnfcl 8190	. . . . . . . . . . . 12 |- ( ph -> ( _E We ( G supp (/) ) /\ dom O e. om ) )
17	16	simpld 466	. . . . . . . . . . 11 |- ( ph -> _E We ( G supp (/) ) )
18	2	oiiso 8070	. . . . . . . . . . 11 |- ( ( ( G supp (/) ) e. _V /\ _E We ( G supp (/) ) ) -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
19	15, 17, 18	syl2anc 673	. . . . . . . . . 10 |- ( ph -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
20		isof1o 6234	. . . . . . . . . 10 |- ( O Isom _E , _E ( dom O , ( G supp (/) ) ) -> O : dom O -1-1-onto-> ( G supp (/) ) )
21	19, 20	syl 17	. . . . . . . . 9 |- ( ph -> O : dom O -1-1-onto-> ( G supp (/) ) )
22		f1ocnv 5840	. . . . . . . . 9 |- ( O : dom O -1-1-onto-> ( G supp (/) ) -> `' O : ( G supp (/) ) -1-1-onto-> dom O )
23		f1of 5828	. . . . . . . . 9 |- ( `' O : ( G supp (/) ) -1-1-onto-> dom O -> `' O : ( G supp (/) ) --> dom O )
24	21, 22, 23	3syl 18	. . . . . . . 8 |- ( ph -> `' O : ( G supp (/) ) --> dom O )
25		oemapval.t	. . . . . . . . 9 |- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) }
26		oemapval.f	. . . . . . . . 9 |- ( ph -> F e. S )
27		oemapvali.r	. . . . . . . . 9 |- ( ph -> F T G )
28		oemapvali.x	. . . . . . . . 9 |- X = U. { c e. B | ( F ` c ) e. ( G ` c ) }
29	7, 8, 4, 25, 26, 6, 27, 28	cantnflem1a 8208	. . . . . . . 8 |- ( ph -> X e. ( G supp (/) ) )
30	24, 29	ffvelrnd 6038	. . . . . . 7 |- ( ph -> ( `' O ` X ) e. dom O )
31		ordelon 5454	. . . . . . 7 |- ( ( Ord dom O /\ ( `' O ` X ) e. dom O ) -> ( `' O ` X ) e. On )
32	3, 30, 31	sylancr 676	. . . . . 6 |- ( ph -> ( `' O ` X ) e. On )
33	32	adantr 472	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( `' O ` X ) e. On )
34	3	a1i 11	. . . . . . . 8 |- ( ph -> Ord dom O )
35		ordelon 5454	. . . . . . . 8 |- ( ( Ord dom O /\ suc u e. dom O ) -> suc u e. On )
36	34, 35	sylan 479	. . . . . . 7 |- ( ( ph /\ suc u e. dom O ) -> suc u e. On )
37		sucelon 6663	. . . . . . 7 |- ( u e. On <-> suc u e. On )
38	36, 37	sylibr 217	. . . . . 6 |- ( ( ph /\ suc u e. dom O ) -> u e. On )
39	38	adantrr 731	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> u e. On )
40		ontri1 5464	. . . . 5 |- ( ( ( `' O ` X ) e. On /\ u e. On ) -> ( ( `' O ` X ) C_ u <-> -. u e. ( `' O ` X ) ) )
41	33, 39, 40	syl2anc 673	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( ( `' O ` X ) C_ u <-> -. u e. ( `' O ` X ) ) )
42	1, 41	mpbid 215	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> -. u e. ( `' O ` X ) )
43	19	adantr 472	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
44		ordtr 5444	. . . . . . . 8 |- ( Ord dom O -> Tr dom O )
45	3, 44	mp1i 13	. . . . . . 7 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> Tr dom O )
46		simprl 772	. . . . . . 7 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> suc u e. dom O )
47		trsuc 5514	. . . . . . 7 |- ( ( Tr dom O /\ suc u e. dom O ) -> u e. dom O )
48	45, 46, 47	syl2anc 673	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> u e. dom O )
49	30	adantr 472	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( `' O ` X ) e. dom O )
50		isorel 6235	. . . . . 6 |- ( ( O Isom _E , _E ( dom O , ( G supp (/) ) ) /\ ( u e. dom O /\ ( `' O ` X ) e. dom O ) ) -> ( u _E ( `' O ` X ) <-> ( O ` u ) _E ( O ` ( `' O ` X ) ) ) )
51	43, 48, 49, 50	syl12anc 1290	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u _E ( `' O ` X ) <-> ( O ` u ) _E ( O ` ( `' O ` X ) ) ) )
52		fvex 5889	. . . . . 6 |- ( `' O ` X ) e. _V
53	52	epelc 4752	. . . . 5 |- ( u _E ( `' O ` X ) <-> u e. ( `' O ` X ) )
54		fvex 5889	. . . . . 6 |- ( O ` ( `' O ` X ) ) e. _V
55	54	epelc 4752	. . . . 5 |- ( ( O ` u ) _E ( O ` ( `' O ` X ) ) <-> ( O ` u ) e. ( O ` ( `' O ` X ) ) )
56	51, 53, 55	3bitr3g 295	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u e. ( `' O ` X ) <-> ( O ` u ) e. ( O ` ( `' O ` X ) ) ) )
57		f1ocnvfv2 6194	. . . . . . 7 |- ( ( O : dom O -1-1-onto-> ( G supp (/) ) /\ X e. ( G supp (/) ) ) -> ( O ` ( `' O ` X ) ) = X )
58	21, 29, 57	syl2anc 673	. . . . . 6 |- ( ph -> ( O ` ( `' O ` X ) ) = X )
59	58	adantr 472	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` ( `' O ` X ) ) = X )
60	59	eleq2d 2534	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( ( O ` u ) e. ( O ` ( `' O ` X ) ) <-> ( O ` u ) e. X ) )
61	56, 60	bitrd 261	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u e. ( `' O ` X ) <-> ( O ` u ) e. X ) )
62	42, 61	mtbid 307	. 2 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> -. ( O ` u ) e. X )
63	7, 8, 4, 25, 26, 6, 27, 28	oemapvali 8207	. . . . . 6 |- ( ph -> ( X e. B /\ ( F ` X ) e. ( G ` X ) /\ A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) ) )
64	63	simp1d 1042	. . . . 5 |- ( ph -> X e. B )
65		onelon 5455	. . . . 5 |- ( ( B e. On /\ X e. B ) -> X e. On )
66	4, 64, 65	syl2anc 673	. . . 4 |- ( ph -> X e. On )
67	66	adantr 472	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X e. On )
68	4	adantr 472	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> B e. On )
69	14	adantr 472	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( G supp (/) ) C_ B )
70	2	oif 8063	. . . . . . 7 |- O : dom O --> ( G supp (/) )
71	70	ffvelrni 6036	. . . . . 6 |- ( u e. dom O -> ( O ` u ) e. ( G supp (/) ) )
72	48, 71	syl 17	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. ( G supp (/) ) )
73	69, 72	sseldd 3419	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. B )
74		onelon 5455	. . . 4 |- ( ( B e. On /\ ( O ` u ) e. B ) -> ( O ` u ) e. On )
75	68, 73, 74	syl2anc 673	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. On )
76		ontri1 5464	. . 3 |- ( ( X e. On /\ ( O ` u ) e. On ) -> ( X C_ ( O ` u ) <-> -. ( O ` u ) e. X ) )
77	67, 75, 76	syl2anc 673	. 2 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( X C_ ( O ` u ) <-> -. ( O ` u ) e. X ) )
78	62, 77	mpbird 240	1 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X C_ ( O ` u ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   C_ wss 3390   (/)c0 3722   U.cuni 4190   class class class wbr 4395   {copab 4453   Tr wtr 4490   _E cep 4748   We wwe 4797   `'ccnv 4838   dom cdm 4839   Ord word 5429   Oncon0 5430   suc csuc 5432   -->wf 5585   -1-1-onto->wf1o 5588   ` cfv 5589   Isom wiso 5590  (class class class)co 6308   omcom 6711   supp csupp 6933   finSupp cfsupp 7901  OrdIsocoi 8042   CNF ccnf 8184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-seqom 7183  df-1o 7200  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-oi 8043  df-cnf 8185

This theorem is referenced by:  cantnflem1c  8210