Theorem cantnflem1b 8188

Description: Lemma for cantnf 8195. (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 765	. . . 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 8041	. . . . . . 7 |- Ord dom O
4		cantnfs.b	. . . . . . . . . . . 12 |- ( ph -> B e. On )
5		suppssdm 6924	. . . . . . . . . . . . 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 8168	. . . . . . . . . . . . . . . 16 |- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) )
10	6, 9	mpbid 214	. . . . . . . . . . . . . . 15 |- ( ph -> ( G : B --> A /\ G finSupp (/) ) )
11	10	simpld 461	. . . . . . . . . . . . . 14 |- ( ph -> G : B --> A )
12		fdm 5731	. . . . . . . . . . . . . 14 |- ( G : B --> A -> dom G = B )
13	11, 12	syl 17	. . . . . . . . . . . . 13 |- ( ph -> dom G = B )
14	5, 13	syl5sseq 3479	. . . . . . . . . . . 12 |- ( ph -> ( G supp (/) ) C_ B )
15	4, 14	ssexd 4549	. . . . . . . . . . 11 |- ( ph -> ( G supp (/) ) e. _V )
16	7, 8, 4, 2, 6	cantnfcl 8169	. . . . . . . . . . . 12 |- ( ph -> ( _E We ( G supp (/) ) /\ dom O e. om ) )
17	16	simpld 461	. . . . . . . . . . 11 |- ( ph -> _E We ( G supp (/) ) )
18	2	oiiso 8049	. . . . . . . . . . 11 |- ( ( ( G supp (/) ) e. _V /\ _E We ( G supp (/) ) ) -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
19	15, 17, 18	syl2anc 666	. . . . . . . . . 10 |- ( ph -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
20		isof1o 6214	. . . . . . . . . 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 5824	. . . . . . . . 9 |- ( O : dom O -1-1-onto-> ( G supp (/) ) -> `' O : ( G supp (/) ) -1-1-onto-> dom O )
23		f1of 5812	. . . . . . . . 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 8187	. . . . . . . 8 |- ( ph -> X e. ( G supp (/) ) )
30	24, 29	ffvelrnd 6021	. . . . . . 7 |- ( ph -> ( `' O ` X ) e. dom O )
31		ordelon 5446	. . . . . . 7 |- ( ( Ord dom O /\ ( `' O ` X ) e. dom O ) -> ( `' O ` X ) e. On )
32	3, 30, 31	sylancr 668	. . . . . 6 |- ( ph -> ( `' O ` X ) e. On )
33	32	adantr 467	. . . . 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 5446	. . . . . . . 8 |- ( ( Ord dom O /\ suc u e. dom O ) -> suc u e. On )
36	34, 35	sylan 474	. . . . . . 7 |- ( ( ph /\ suc u e. dom O ) -> suc u e. On )
37		sucelon 6641	. . . . . . 7 |- ( u e. On <-> suc u e. On )
38	36, 37	sylibr 216	. . . . . 6 |- ( ( ph /\ suc u e. dom O ) -> u e. On )
39	38	adantrr 722	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> u e. On )
40		ontri1 5456	. . . . 5 |- ( ( ( `' O ` X ) e. On /\ u e. On ) -> ( ( `' O ` X ) C_ u <-> -. u e. ( `' O ` X ) ) )
41	33, 39, 40	syl2anc 666	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( ( `' O ` X ) C_ u <-> -. u e. ( `' O ` X ) ) )
42	1, 41	mpbid 214	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> -. u e. ( `' O ` X ) )
43	19	adantr 467	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> O Isom _E , _E ( dom O , ( G supp (/) ) ) )
44		ordtr 5436	. . . . . . . 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 763	. . . . . . 7 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> suc u e. dom O )
47		trsuc 5506	. . . . . . 7 |- ( ( Tr dom O /\ suc u e. dom O ) -> u e. dom O )
48	45, 46, 47	syl2anc 666	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> u e. dom O )
49	30	adantr 467	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( `' O ` X ) e. dom O )
50		isorel 6215	. . . . . 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 1265	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u _E ( `' O ` X ) <-> ( O ` u ) _E ( O ` ( `' O ` X ) ) ) )
52		fvex 5873	. . . . . 6 |- ( `' O ` X ) e. _V
53	52	epelc 4746	. . . . 5 |- ( u _E ( `' O ` X ) <-> u e. ( `' O ` X ) )
54		fvex 5873	. . . . . 6 |- ( O ` ( `' O ` X ) ) e. _V
55	54	epelc 4746	. . . . 5 |- ( ( O ` u ) _E ( O ` ( `' O ` X ) ) <-> ( O ` u ) e. ( O ` ( `' O ` X ) ) )
56	51, 53, 55	3bitr3g 291	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u e. ( `' O ` X ) <-> ( O ` u ) e. ( O ` ( `' O ` X ) ) ) )
57		f1ocnvfv2 6174	. . . . . . 7 |- ( ( O : dom O -1-1-onto-> ( G supp (/) ) /\ X e. ( G supp (/) ) ) -> ( O ` ( `' O ` X ) ) = X )
58	21, 29, 57	syl2anc 666	. . . . . 6 |- ( ph -> ( O ` ( `' O ` X ) ) = X )
59	58	adantr 467	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` ( `' O ` X ) ) = X )
60	59	eleq2d 2513	. . . 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 257	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( u e. ( `' O ` X ) <-> ( O ` u ) e. X ) )
62	42, 61	mtbid 302	. 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 8186	. . . . . 6 |- ( ph -> ( X e. B /\ ( F ` X ) e. ( G ` X ) /\ A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) ) )
64	63	simp1d 1019	. . . . 5 |- ( ph -> X e. B )
65		onelon 5447	. . . . 5 |- ( ( B e. On /\ X e. B ) -> X e. On )
66	4, 64, 65	syl2anc 666	. . . 4 |- ( ph -> X e. On )
67	66	adantr 467	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X e. On )
68	4	adantr 467	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> B e. On )
69	14	adantr 467	. . . . 5 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( G supp (/) ) C_ B )
70	2	oif 8042	. . . . . . 7 |- O : dom O --> ( G supp (/) )
71	70	ffvelrni 6019	. . . . . 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 3432	. . . 4 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. B )
74		onelon 5447	. . . 4 |- ( ( B e. On /\ ( O ` u ) e. B ) -> ( O ` u ) e. On )
75	68, 73, 74	syl2anc 666	. . 3 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( O ` u ) e. On )
76		ontri1 5456	. . 3 |- ( ( X e. On /\ ( O ` u ) e. On ) -> ( X C_ ( O ` u ) <-> -. ( O ` u ) e. X ) )
77	67, 75, 76	syl2anc 666	. 2 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> ( X C_ ( O ` u ) <-> -. ( O ` u ) e. X ) )
78	62, 77	mpbird 236	1 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X C_ ( O ` u ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 3044   C_ wss 3403   (/)c0 3730   U.cuni 4197   class class class wbr 4401   {copab 4459   Tr wtr 4496   _E cep 4742   We wwe 4791   `'ccnv 4832   dom cdm 4833   Ord word 5421   Oncon0 5422   suc csuc 5424   -->wf 5577   -1-1-onto->wf1o 5580   ` cfv 5581   Isom wiso 5582  (class class class)co 6288   omcom 6689   supp csupp 6911   finSupp cfsupp 7880  OrdIsocoi 8021   CNF ccnf 8163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-fal 1449  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-seqom 7162  df-1o 7179  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-oi 8022  df-cnf 8164

This theorem is referenced by:  cantnflem1c  8189