Theorem cantnflem1b 8122

Description: Lemma for cantnf 8129. (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 757	. . . 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 7972	. . . . . . 7 |- Ord dom O
4		cantnfs.b	. . . . . . . . . . . 12 |- ( ph -> B e. On )
5		suppssdm 6930	. . . . . . . . . . . . 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 8102	. . . . . . . . . . . . . . . 16 |- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) )
10	6, 9	mpbid 210	. . . . . . . . . . . . . . 15 |- ( ph -> ( G : B --> A /\ G finSupp (/) ) )
11	10	simpld 459	. . . . . . . . . . . . . 14 |- ( ph -> G : B --> A )
12		fdm 5741	. . . . . . . . . . . . . 14 |- ( G : B --> A -> dom G = B )
13	11, 12	syl 16	. . . . . . . . . . . . 13 |- ( ph -> dom G = B )
14	5, 13	syl5sseq 3547	. . . . . . . . . . . 12 |- ( ph -> ( G supp (/) ) C_ B )
15	4, 14	ssexd 4603	. . . . . . . . . . 11 |- ( ph -> ( G supp (/) ) e. _V )
16	7, 8, 4, 2, 6	cantnfcl 8103	. . . . . . . . . . . 12 |- ( ph -> ( _E We ( G supp (/) ) /\ dom O e. om ) )
17	16	simpld 459	. . . . . . . . . . 11 |- ( ph -> _E We ( G supp (/) ) )
18	2	oiiso 7980	. . . . . . . . . . 11 |- ( ( ( G supp (/) ) e. _V /\ _E We ( G supp (/) ) ) -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
19	15, 17, 18	syl2anc 661	. . . . . . . . . 10 |- ( ph -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
20		isof1o 6222	. . . . . . . . . 10 |- ( O Isom _E , _E ( dom O , ( G supp (/) ) ) -> O : dom O -1-1-onto-> ( G supp (/) ) )
21	19, 20	syl 16	. . . . . . . . 9 |- ( ph -> O : dom O -1-1-onto-> ( G supp (/) ) )
22		f1ocnv 5834	. . . . . . . . 9 |- ( O : dom O -1-1-onto-> ( G supp (/) ) -> `' O : ( G supp (/) ) -1-1-onto-> dom O )
23		f1of 5822	. . . . . . . . 9 |- ( `' O : ( G supp (/) ) -1-1-onto-> dom O -> `' O : ( G supp (/) ) --> dom O )
24	21, 22, 23	3syl 20	. . . . . . . 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 8121	. . . . . . . 8 |- ( ph -> X e. ( G supp (/) ) )
30	24, 29	ffvelrnd 6033	. . . . . . 7 |- ( ph -> ( `' O ` X ) e. dom O )
31		ordelon 4911	. . . . . . 7 |- ( ( Ord dom O /\ ( `' O ` X ) e. dom O ) -> ( `' O ` X ) e. On )
32	3, 30, 31	sylancr 663	. . . . . 6 |- ( ph -> ( `' O ` X ) e. On )
33	32	adantr 465	. . . . 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 4911	. . . . . . . 8 |- ( ( Ord dom O /\ suc u e. dom O ) -> suc u e. On )
36	34, 35	sylan 471	. . . . . . 7 |- ( ( ph /\ suc u e. dom O ) -> suc u e. On )
37		sucelon 6651	. . . . . . 7 |- ( u e. On <-> suc u e. On )
38	36, 37	sylibr 212	. . . . . 6 |- ( ( ph /\ suc u e. dom O ) -> u e. On )
39	38	adantrr 716	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> u e. On )
40		ontri1 4921	. . . . 5 |- ( ( ( `' O ` X ) e. On /\ u e. On ) -> ( ( `' O ` X ) C_ u <-> -. u e. ( `' O ` X ) ) )
41	33, 39, 40	syl2anc 661	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( ( `' O ` X ) C_ u <-> -. u e. ( `' O ` X ) ) )
42	1, 41	mpbid 210	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> -. u e. ( `' O ` X ) )
43	19	adantr 465	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
44		ordtr 4901	. . . . . . . 8 |- ( Ord dom O -> Tr dom O )
45	3, 44	mp1i 12	. . . . . . 7 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> Tr dom O )
46		simprl 756	. . . . . . 7 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> suc u e. dom O )
47		trsuc 4971	. . . . . . 7 |- ( ( Tr dom O /\ suc u e. dom O ) -> u e. dom O )
48	45, 46, 47	syl2anc 661	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> u e. dom O )
49	30	adantr 465	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( `' O ` X ) e. dom O )
50		isorel 6223	. . . . . 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 1226	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u _E ( `' O ` X ) <-> ( O ` u ) _E ( O ` ( `' O ` X ) ) ) )
52		fvex 5882	. . . . . 6 |- ( `' O ` X ) e. _V
53	52	epelc 4802	. . . . 5 |- ( u _E ( `' O ` X ) <-> u e. ( `' O ` X ) )
54		fvex 5882	. . . . . 6 |- ( O ` ( `' O ` X ) ) e. _V
55	54	epelc 4802	. . . . 5 |- ( ( O ` u ) _E ( O ` ( `' O ` X ) ) <-> ( O ` u ) e. ( O ` ( `' O ` X ) ) )
56	51, 53, 55	3bitr3g 287	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u e. ( `' O ` X ) <-> ( O ` u ) e. ( O ` ( `' O ` X ) ) ) )
57		f1ocnvfv2 6184	. . . . . . 7 |- ( ( O : dom O -1-1-onto-> ( G supp (/) ) /\ X e. ( G supp (/) ) ) -> ( O ` ( `' O ` X ) ) = X )
58	21, 29, 57	syl2anc 661	. . . . . 6 |- ( ph -> ( O ` ( `' O ` X ) ) = X )
59	58	adantr 465	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` ( `' O ` X ) ) = X )
60	59	eleq2d 2527	. . . 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 253	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u e. ( `' O ` X ) <-> ( O ` u ) e. X ) )
62	42, 61	mtbid 300	. 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 8120	. . . . . 6 |- ( ph -> ( X e. B /\ ( F ` X ) e. ( G ` X ) /\ A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) ) )
64	63	simp1d 1008	. . . . 5 |- ( ph -> X e. B )
65		onelon 4912	. . . . 5 |- ( ( B e. On /\ X e. B ) -> X e. On )
66	4, 64, 65	syl2anc 661	. . . 4 |- ( ph -> X e. On )
67	66	adantr 465	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X e. On )
68	4	adantr 465	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> B e. On )
69	14	adantr 465	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( G supp (/) ) C_ B )
70	2	oif 7973	. . . . . . 7 |- O : dom O --> ( G supp (/) )
71	70	ffvelrni 6031	. . . . . 6 |- ( u e. dom O -> ( O ` u ) e. ( G supp (/) ) )
72	48, 71	syl 16	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. ( G supp (/) ) )
73	69, 72	sseldd 3500	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. B )
74		onelon 4912	. . . 4 |- ( ( B e. On /\ ( O ` u ) e. B ) -> ( O ` u ) e. On )
75	68, 73, 74	syl2anc 661	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. On )
76		ontri1 4921	. . 3 |- ( ( X e. On /\ ( O ` u ) e. On ) -> ( X C_ ( O ` u ) <-> -. ( O ` u ) e. X ) )
77	67, 75, 76	syl2anc 661	. 2 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( X C_ ( O ` u ) <-> -. ( O ` u ) e. X ) )
78	62, 77	mpbird 232	1 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X C_ ( O ` u ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3109   C_ wss 3471   (/)c0 3793   U.cuni 4251   class class class wbr 4456   {copab 4514   Tr wtr 4550   _E cep 4798   We wwe 4846   Ord word 4886   Oncon0 4887   suc csuc 4889   `'ccnv 5007   dom cdm 5008   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594   Isom wiso 5595  (class class class)co 6296   omcom 6699   supp csupp 6917   finSupp cfsupp 7847  OrdIsocoi 7952   CNF ccnf 8095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-cnf 8096

This theorem is referenced by:  cantnflem1c  8123