Theorem cantnflem1b 8006

Description: Lemma for cantnf 8013. (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 756	. . . 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 7855	. . . . . . 7 |- Ord dom O
4		cantnfs.b	. . . . . . . . . . . 12 |- ( ph -> B e. On )
5		suppssdm 6814	. . . . . . . . . . . . 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 7986	. . . . . . . . . . . . . . . 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 5672	. . . . . . . . . . . . . 14 |- ( G : B --> A -> dom G = B )
13	11, 12	syl 16	. . . . . . . . . . . . 13 |- ( ph -> dom G = B )
14	5, 13	syl5sseq 3513	. . . . . . . . . . . 12 |- ( ph -> ( G supp (/) ) C_ B )
15	4, 14	ssexd 4548	. . . . . . . . . . 11 |- ( ph -> ( G supp (/) ) e. _V )
16	7, 8, 4, 2, 6	cantnfcl 7987	. . . . . . . . . . . 12 |- ( ph -> ( _E We ( G supp (/) ) /\ dom O e. om ) )
17	16	simpld 459	. . . . . . . . . . 11 |- ( ph -> _E We ( G supp (/) ) )
18	2	oiiso 7863	. . . . . . . . . . 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 6126	. . . . . . . . . 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 5762	. . . . . . . . 9 |- ( O : dom O -1-1-onto-> ( G supp (/) ) -> `' O : ( G supp (/) ) -1-1-onto-> dom O )
23		f1of 5750	. . . . . . . . 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 8005	. . . . . . . 8 |- ( ph -> X e. ( G supp (/) ) )
30	24, 29	ffvelrnd 5954	. . . . . . 7 |- ( ph -> ( `' O ` X ) e. dom O )
31		ordelon 4852	. . . . . . 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 4852	. . . . . . . 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 6539	. . . . . . 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 4862	. . . . 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 4842	. . . . . . . 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 755	. . . . . . 7 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> suc u e. dom O )
47		trsuc 4912	. . . . . . 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 6127	. . . . . 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 1217	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u _E ( `' O ` X ) <-> ( O ` u ) _E ( O ` ( `' O ` X ) ) ) )
52		fvex 5810	. . . . . 6 |- ( `' O ` X ) e. _V
53	52	epelc 4743	. . . . 5 |- ( u _E ( `' O ` X ) <-> u e. ( `' O ` X ) )
54		fvex 5810	. . . . . 6 |- ( O ` ( `' O ` X ) ) e. _V
55	54	epelc 4743	. . . . 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 6094	. . . . . . 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 2524	. . . 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 8004	. . . . . 6 |- ( ph -> ( X e. B /\ ( F ` X ) e. ( G ` X ) /\ A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) ) )
64	63	simp1d 1000	. . . . 5 |- ( ph -> X e. B )
65		onelon 4853	. . . . 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 7856	. . . . . . 7 |- O : dom O --> ( G supp (/) )
71	70	ffvelrni 5952	. . . . . 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 3466	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. B )
74		onelon 4853	. . . 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 4862	. . 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 1370   e. wcel 1758   A.wral 2799   E.wrex 2800   {crab 2803   _Vcvv 3078   C_ wss 3437   (/)c0 3746   U.cuni 4200   class class class wbr 4401   {copab 4458   Tr wtr 4494   _E cep 4739   We wwe 4787   Ord word 4827   Oncon0 4828   suc csuc 4830   `'ccnv 4948   dom cdm 4949   -->wf 5523   -1-1-onto->wf1o 5526   ` cfv 5527   Isom wiso 5528  (class class class)co 6201   omcom 6587   supp csupp 6801   finSupp cfsupp 7732  OrdIsocoi 7835   CNF ccnf 7979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-supp 6802  df-recs 6943  df-rdg 6977  df-seqom 7014  df-1o 7031  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-oi 7836  df-cnf 7980

This theorem is referenced by:  cantnflem1c  8007