Theorem cantnflem1b 8583

Description: Lemma for cantnf 8590. (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 796	. . . 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 8434	. . . . . . 7 |- Ord dom O
4		cantnfs.b	. . . . . . . . . . . 12 |- ( ph -> B e. On )
5		suppssdm 7308	. . . . . . . . . . . . 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 8563	. . . . . . . . . . . . . . . 16 |- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) )
10	6, 9	mpbid 222	. . . . . . . . . . . . . . 15 |- ( ph -> ( G : B --> A /\ G finSupp (/) ) )
11	10	simpld 475	. . . . . . . . . . . . . 14 |- ( ph -> G : B --> A )
12		fdm 6051	. . . . . . . . . . . . . 14 |- ( G : B --> A -> dom G = B )
13	11, 12	syl 17	. . . . . . . . . . . . 13 |- ( ph -> dom G = B )
14	5, 13	syl5sseq 3653	. . . . . . . . . . . 12 |- ( ph -> ( G supp (/) ) C_ B )
15	4, 14	ssexd 4805	. . . . . . . . . . 11 |- ( ph -> ( G supp (/) ) e. _V )
16	7, 8, 4, 2, 6	cantnfcl 8564	. . . . . . . . . . . 12 |- ( ph -> ( _E We ( G supp (/) ) /\ dom O e. om ) )
17	16	simpld 475	. . . . . . . . . . 11 |- ( ph -> _E We ( G supp (/) ) )
18	2	oiiso 8442	. . . . . . . . . . 11 |- ( ( ( G supp (/) ) e. _V /\ _E We ( G supp (/) ) ) -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
19	15, 17, 18	syl2anc 693	. . . . . . . . . 10 |- ( ph -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
20		isof1o 6573	. . . . . . . . . 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 6149	. . . . . . . . 9 |- ( O : dom O -1-1-onto-> ( G supp (/) ) -> `' O : ( G supp (/) ) -1-1-onto-> dom O )
23		f1of 6137	. . . . . . . . 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 8582	. . . . . . . 8 |- ( ph -> X e. ( G supp (/) ) )
30	24, 29	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( `' O ` X ) e. dom O )
31		ordelon 5747	. . . . . . 7 |- ( ( Ord dom O /\ ( `' O ` X ) e. dom O ) -> ( `' O ` X ) e. On )
32	3, 30, 31	sylancr 695	. . . . . 6 |- ( ph -> ( `' O ` X ) e. On )
33	32	adantr 481	. . . . 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 5747	. . . . . . . 8 |- ( ( Ord dom O /\ suc u e. dom O ) -> suc u e. On )
36	34, 35	sylan 488	. . . . . . 7 |- ( ( ph /\ suc u e. dom O ) -> suc u e. On )
37		sucelon 7017	. . . . . . 7 |- ( u e. On <-> suc u e. On )
38	36, 37	sylibr 224	. . . . . 6 |- ( ( ph /\ suc u e. dom O ) -> u e. On )
39	38	adantrr 753	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> u e. On )
40		ontri1 5757	. . . . 5 |- ( ( ( `' O ` X ) e. On /\ u e. On ) -> ( ( `' O ` X ) C_ u <-> -. u e. ( `' O ` X ) ) )
41	33, 39, 40	syl2anc 693	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( ( `' O ` X ) C_ u <-> -. u e. ( `' O ` X ) ) )
42	1, 41	mpbid 222	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> -. u e. ( `' O ` X ) )
43	19	adantr 481	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
44		ordtr 5737	. . . . . . . 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 794	. . . . . . 7 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> suc u e. dom O )
47		trsuc 5810	. . . . . . 7 |- ( ( Tr dom O /\ suc u e. dom O ) -> u e. dom O )
48	45, 46, 47	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> u e. dom O )
49	30	adantr 481	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( `' O ` X ) e. dom O )
50		isorel 6576	. . . . . 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 1324	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u _E ( `' O ` X ) <-> ( O ` u ) _E ( O ` ( `' O ` X ) ) ) )
52		fvex 6201	. . . . . 6 |- ( `' O ` X ) e. _V
53	52	epelc 5031	. . . . 5 |- ( u _E ( `' O ` X ) <-> u e. ( `' O ` X ) )
54		fvex 6201	. . . . . 6 |- ( O ` ( `' O ` X ) ) e. _V
55	54	epelc 5031	. . . . 5 |- ( ( O ` u ) _E ( O ` ( `' O ` X ) ) <-> ( O ` u ) e. ( O ` ( `' O ` X ) ) )
56	51, 53, 55	3bitr3g 302	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u e. ( `' O ` X ) <-> ( O ` u ) e. ( O ` ( `' O ` X ) ) ) )
57		f1ocnvfv2 6533	. . . . . . 7 |- ( ( O : dom O -1-1-onto-> ( G supp (/) ) /\ X e. ( G supp (/) ) ) -> ( O ` ( `' O ` X ) ) = X )
58	21, 29, 57	syl2anc 693	. . . . . 6 |- ( ph -> ( O ` ( `' O ` X ) ) = X )
59	58	adantr 481	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` ( `' O ` X ) ) = X )
60	59	eleq2d 2687	. . . 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 268	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u e. ( `' O ` X ) <-> ( O ` u ) e. X ) )
62	42, 61	mtbid 314	. 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 8581	. . . . . 6 |- ( ph -> ( X e. B /\ ( F ` X ) e. ( G ` X ) /\ A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) ) )
64	63	simp1d 1073	. . . . 5 |- ( ph -> X e. B )
65		onelon 5748	. . . . 5 |- ( ( B e. On /\ X e. B ) -> X e. On )
66	4, 64, 65	syl2anc 693	. . . 4 |- ( ph -> X e. On )
67	66	adantr 481	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X e. On )
68	4	adantr 481	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> B e. On )
69	14	adantr 481	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( G supp (/) ) C_ B )
70	2	oif 8435	. . . . . . 7 |- O : dom O --> ( G supp (/) )
71	70	ffvelrni 6358	. . . . . 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 3604	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. B )
74		onelon 5748	. . . 4 |- ( ( B e. On /\ ( O ` u ) e. B ) -> ( O ` u ) e. On )
75	68, 73, 74	syl2anc 693	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. On )
76		ontri1 5757	. . 3 |- ( ( X e. On /\ ( O ` u ) e. On ) -> ( X C_ ( O ` u ) <-> -. ( O ` u ) e. X ) )
77	67, 75, 76	syl2anc 693	. 2 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( X C_ ( O ` u ) <-> -. ( O ` u ) e. X ) )
78	62, 77	mpbird 247	1 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X C_ ( O ` u ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U.cuni 4436   class class class wbr 4653   {copab 4712   Tr wtr 4752   _E cep 5028   We wwe 5072   `'ccnv 5113   dom cdm 5114   Ord word 5722   Oncon0 5723   suc csuc 5725   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   omcom 7065   supp csupp 7295   finSupp cfsupp 8275  OrdIsocoi 8414   CNF ccnf 8558

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-fal 1489  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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-cnf 8559

This theorem is referenced by:  cantnflem1c  8584