Theorem cantnflem1c 8584

Description: Lemma for cantnf 8590. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.)

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
cantnflem1c	|- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> x e. ( G supp (/) ) )

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 cantnflem1c

Step	Hyp	Ref	Expression
1		cantnfs.b	. . 3 |- ( ph -> B e. On )
2	1	ad3antrrr 766	. 2 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> B e. On )
3		simplr 792	. 2 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> x e. B )
4		oemapval.g	. . . . . 6 |- ( ph -> G e. S )
5		cantnfs.s	. . . . . . 7 |- S = dom ( A CNF B )
6		cantnfs.a	. . . . . . 7 |- ( ph -> A e. On )
7	5, 6, 1	cantnfs 8563	. . . . . 6 |- ( ph -> ( G e. S <-> ( G : B --> A /\ G finSupp (/) ) ) )
8	4, 7	mpbid 222	. . . . 5 |- ( ph -> ( G : B --> A /\ G finSupp (/) ) )
9	8	simpld 475	. . . 4 |- ( ph -> G : B --> A )
10		ffn 6045	. . . 4 |- ( G : B --> A -> G Fn B )
11	9, 10	syl 17	. . 3 |- ( ph -> G Fn B )
12	11	ad3antrrr 766	. 2 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> G Fn B )
13		oemapval.t	. . . . . . 7 |- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) }
14		oemapval.f	. . . . . . 7 |- ( ph -> F e. S )
15		oemapvali.r	. . . . . . 7 |- ( ph -> F T G )
16		oemapvali.x	. . . . . . 7 |- X = U. { c e. B | ( F ` c ) e. ( G ` c ) }
17	5, 6, 1, 13, 14, 4, 15, 16	oemapvali 8581	. . . . . 6 |- ( ph -> ( X e. B /\ ( F ` X ) e. ( G ` X ) /\ A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) ) )
18	17	simp3d 1075	. . . . 5 |- ( ph -> A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) )
19	18	ad3antrrr 766	. . . 4 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) )
20		cantnflem1.o	. . . . . . 7 |- O = OrdIso ( _E , ( G supp (/) ) )
21	5, 6, 1, 13, 14, 4, 15, 16, 20	cantnflem1b 8583	. . . . . 6 |- ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) -> X C_ ( O ` u ) )
22	21	ad2antrr 762	. . . . 5 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> X C_ ( O ` u ) )
23		simprr 796	. . . . 5 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> ( O ` u ) e. x )
24	17	simp1d 1073	. . . . . . . 8 |- ( ph -> X e. B )
25		onelon 5748	. . . . . . . 8 |- ( ( B e. On /\ X e. B ) -> X e. On )
26	1, 24, 25	syl2anc 693	. . . . . . 7 |- ( ph -> X e. On )
27	26	ad3antrrr 766	. . . . . 6 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> X e. On )
28		onss 6990	. . . . . . . . . 10 |- ( B e. On -> B C_ On )
29	1, 28	syl 17	. . . . . . . . 9 |- ( ph -> B C_ On )
30	29	sselda 3603	. . . . . . . 8 |- ( ( ph /\ x e. B ) -> x e. On )
31	30	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) -> x e. On )
32	31	adantr 481	. . . . . 6 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> x e. On )
33		ontr2 5772	. . . . . 6 |- ( ( X e. On /\ x e. On ) -> ( ( X C_ ( O ` u ) /\ ( O ` u ) e. x ) -> X e. x ) )
34	27, 32, 33	syl2anc 693	. . . . 5 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> ( ( X C_ ( O ` u ) /\ ( O ` u ) e. x ) -> X e. x ) )
35	22, 23, 34	mp2and 715	. . . 4 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> X e. x )
36		eleq2 2690	. . . . . 6 |- ( w = x -> ( X e. w <-> X e. x ) )
37		fveq2 6191	. . . . . . 7 |- ( w = x -> ( F ` w ) = ( F ` x ) )
38		fveq2 6191	. . . . . . 7 |- ( w = x -> ( G ` w ) = ( G ` x ) )
39	37, 38	eqeq12d 2637	. . . . . 6 |- ( w = x -> ( ( F ` w ) = ( G ` w ) <-> ( F ` x ) = ( G ` x ) ) )
40	36, 39	imbi12d 334	. . . . 5 |- ( w = x -> ( ( X e. w -> ( F ` w ) = ( G ` w ) ) <-> ( X e. x -> ( F ` x ) = ( G ` x ) ) ) )
41	40	rspcv 3305	. . . 4 |- ( x e. B -> ( A. w e. B ( X e. w -> ( F ` w ) = ( G ` w ) ) -> ( X e. x -> ( F ` x ) = ( G ` x ) ) ) )
42	3, 19, 35, 41	syl3c 66	. . 3 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> ( F ` x ) = ( G ` x ) )
43		simprl 794	. . 3 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> ( F ` x ) =/= (/) )
44	42, 43	eqnetrrd 2862	. 2 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> ( G ` x ) =/= (/) )
45		fvn0elsupp 7311	. 2 |- ( ( ( B e. On /\ x e. B ) /\ ( G Fn B /\ ( G ` x ) =/= (/) ) ) -> x e. ( G supp (/) ) )
46	2, 3, 12, 44, 45	syl22anc 1327	1 |- ( ( ( ( ph /\ ( suc u e. dom O /\ ( `' O ` X ) C_ u ) ) /\ x e. B ) /\ ( ( F ` x ) =/= (/) /\ ( O ` u ) e. x ) ) -> x e. ( G supp (/) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   U.cuni 4436   class class class wbr 4653   {copab 4712   _E cep 5028   `'ccnv 5113   dom cdm 5114   Oncon0 5723   suc csuc 5725   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   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:  cantnflem1  8586