Theorem cantnflem4 8121

Description: Lemma for cantnf 8122. Complete the induction step of cantnflem3 8120. (Contributed by Mario Carneiro, 25-May-2015.)

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 ) ) ) }
cantnf.c	|- ( ph -> C e. ( A ^o B ) )
cantnf.s	|- ( ph -> C C_ ran ( A CNF B ) )
cantnf.e	|- ( ph -> (/) e. C )
cantnf.x	|- X = U. |^| { c e. On | C e. ( A ^o c ) }
cantnf.p	|- P = ( iota d E. a e. On E. b e. ( A ^o X ) ( d = <. a , b >. /\ ( ( ( A ^o X ) .o a ) +o b ) = C ) )
cantnf.y	|- Y = ( 1st ` P )
cantnf.z	|- Z = ( 2nd ` P )

Assertion

Ref	Expression
cantnflem4	|- ( ph -> C e. ran ( A CNF B ) )

Distinct variable groups:   w, c, x, y, z, B   a, b, c, d, w, x, y, z, C   A, a, b, c, d, w, x, y, z   T, c   S, c, x, y, z   x, Z, y, z   ph, x, y, z   w, Y, x, y, z   X, a, b, d, w, x, y, z

Allowed substitution hints:   ph( w, a, b, c, d)   B( a, b, d)   P( x, y, z, w, a, b, c, d)   S( w, a, b, d)   T( x, y, z, w, a, b, d)   X( c)   Y( a, b, c, d)   Z( w, a, b, c, d)

Proof of Theorem cantnflem4

Dummy variables g t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnf.s	. . . 4 |- ( ph -> C C_ ran ( A CNF B ) )
2		cantnfs.a	. . . . . . . . 9 |- ( ph -> A e. On )
3		cantnfs.s	. . . . . . . . . . . . 13 |- S = dom ( A CNF B )
4		cantnfs.b	. . . . . . . . . . . . 13 |- ( ph -> B e. On )
5		oemapval.t	. . . . . . . . . . . . 13 |- T = { <. x , y >. | E. z e. B ( ( x ` z ) e. ( y ` z ) /\ A. w e. B ( z e. w -> ( x ` w ) = ( y ` w ) ) ) }
6		cantnf.c	. . . . . . . . . . . . 13 |- ( ph -> C e. ( A ^o B ) )
7		cantnf.e	. . . . . . . . . . . . 13 |- ( ph -> (/) e. C )
8	3, 2, 4, 5, 6, 1, 7	cantnflem2 8119	. . . . . . . . . . . 12 |- ( ph -> ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) )
9		eqid 2467	. . . . . . . . . . . . . 14 |- X = X
10		eqid 2467	. . . . . . . . . . . . . 14 |- Y = Y
11		eqid 2467	. . . . . . . . . . . . . 14 |- Z = Z
12	9, 10, 11	3pm3.2i 1174	. . . . . . . . . . . . 13 |- ( X = X /\ Y = Y /\ Z = Z )
13		cantnf.x	. . . . . . . . . . . . . 14 |- X = U. |^| { c e. On | C e. ( A ^o c ) }
14		cantnf.p	. . . . . . . . . . . . . 14 |- P = ( iota d E. a e. On E. b e. ( A ^o X ) ( d = <. a , b >. /\ ( ( ( A ^o X ) .o a ) +o b ) = C ) )
15		cantnf.y	. . . . . . . . . . . . . 14 |- Y = ( 1st ` P )
16		cantnf.z	. . . . . . . . . . . . . 14 |- Z = ( 2nd ` P )
17	13, 14, 15, 16	oeeui 7261	. . . . . . . . . . . . 13 |- ( ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) -> ( ( ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) /\ ( ( ( A ^o X ) .o Y ) +o Z ) = C ) <-> ( X = X /\ Y = Y /\ Z = Z ) ) )
18	12, 17	mpbiri 233	. . . . . . . . . . . 12 |- ( ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) -> ( ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) /\ ( ( ( A ^o X ) .o Y ) +o Z ) = C ) )
19	8, 18	syl 16	. . . . . . . . . . 11 |- ( ph -> ( ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) /\ ( ( ( A ^o X ) .o Y ) +o Z ) = C ) )
20	19	simpld 459	. . . . . . . . . 10 |- ( ph -> ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) )
21	20	simp1d 1008	. . . . . . . . 9 |- ( ph -> X e. On )
22		oecl 7197	. . . . . . . . 9 |- ( ( A e. On /\ X e. On ) -> ( A ^o X ) e. On )
23	2, 21, 22	syl2anc 661	. . . . . . . 8 |- ( ph -> ( A ^o X ) e. On )
24	20	simp2d 1009	. . . . . . . . . 10 |- ( ph -> Y e. ( A \ 1o ) )
25	24	eldifad 3493	. . . . . . . . 9 |- ( ph -> Y e. A )
26		onelon 4908	. . . . . . . . 9 |- ( ( A e. On /\ Y e. A ) -> Y e. On )
27	2, 25, 26	syl2anc 661	. . . . . . . 8 |- ( ph -> Y e. On )
28		omcl 7196	. . . . . . . 8 |- ( ( ( A ^o X ) e. On /\ Y e. On ) -> ( ( A ^o X ) .o Y ) e. On )
29	23, 27, 28	syl2anc 661	. . . . . . 7 |- ( ph -> ( ( A ^o X ) .o Y ) e. On )
30	20	simp3d 1010	. . . . . . . 8 |- ( ph -> Z e. ( A ^o X ) )
31		onelon 4908	. . . . . . . 8 |- ( ( ( A ^o X ) e. On /\ Z e. ( A ^o X ) ) -> Z e. On )
32	23, 30, 31	syl2anc 661	. . . . . . 7 |- ( ph -> Z e. On )
33		oaword1 7211	. . . . . . 7 |- ( ( ( ( A ^o X ) .o Y ) e. On /\ Z e. On ) -> ( ( A ^o X ) .o Y ) C_ ( ( ( A ^o X ) .o Y ) +o Z ) )
34	29, 32, 33	syl2anc 661	. . . . . 6 |- ( ph -> ( ( A ^o X ) .o Y ) C_ ( ( ( A ^o X ) .o Y ) +o Z ) )
35		dif1o 7160	. . . . . . . . . . 11 |- ( Y e. ( A \ 1o ) <-> ( Y e. A /\ Y =/= (/) ) )
36	35	simprbi 464	. . . . . . . . . 10 |- ( Y e. ( A \ 1o ) -> Y =/= (/) )
37	24, 36	syl 16	. . . . . . . . 9 |- ( ph -> Y =/= (/) )
38		on0eln0 4938	. . . . . . . . . 10 |- ( Y e. On -> ( (/) e. Y <-> Y =/= (/) ) )
39	27, 38	syl 16	. . . . . . . . 9 |- ( ph -> ( (/) e. Y <-> Y =/= (/) ) )
40	37, 39	mpbird 232	. . . . . . . 8 |- ( ph -> (/) e. Y )
41		omword1 7232	. . . . . . . 8 |- ( ( ( ( A ^o X ) e. On /\ Y e. On ) /\ (/) e. Y ) -> ( A ^o X ) C_ ( ( A ^o X ) .o Y ) )
42	23, 27, 40, 41	syl21anc 1227	. . . . . . 7 |- ( ph -> ( A ^o X ) C_ ( ( A ^o X ) .o Y ) )
43	42, 30	sseldd 3510	. . . . . 6 |- ( ph -> Z e. ( ( A ^o X ) .o Y ) )
44	34, 43	sseldd 3510	. . . . 5 |- ( ph -> Z e. ( ( ( A ^o X ) .o Y ) +o Z ) )
45	19	simprd 463	. . . . 5 |- ( ph -> ( ( ( A ^o X ) .o Y ) +o Z ) = C )
46	44, 45	eleqtrd 2557	. . . 4 |- ( ph -> Z e. C )
47	1, 46	sseldd 3510	. . 3 |- ( ph -> Z e. ran ( A CNF B ) )
48	3, 2, 4	cantnff 8103	. . . 4 |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )
49		ffn 5736	. . . 4 |- ( ( A CNF B ) : S --> ( A ^o B ) -> ( A CNF B ) Fn S )
50		fvelrnb 5920	. . . 4 |- ( ( A CNF B ) Fn S -> ( Z e. ran ( A CNF B ) <-> E. g e. S ( ( A CNF B ) ` g ) = Z ) )
51	48, 49, 50	3syl 20	. . 3 |- ( ph -> ( Z e. ran ( A CNF B ) <-> E. g e. S ( ( A CNF B ) ` g ) = Z ) )
52	47, 51	mpbid 210	. 2 |- ( ph -> E. g e. S ( ( A CNF B ) ` g ) = Z )
53	2	adantr 465	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> A e. On )
54	4	adantr 465	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> B e. On )
55	6	adantr 465	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C e. ( A ^o B ) )
56	1	adantr 465	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C C_ ran ( A CNF B ) )
57	7	adantr 465	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> (/) e. C )
58		simprl 755	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> g e. S )
59		simprr 756	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> ( ( A CNF B ) ` g ) = Z )
60		eqid 2467	. . 3 |- ( t e. B |-> if ( t = X , Y , ( g ` t ) ) ) = ( t e. B |-> if ( t = X , Y , ( g ` t ) ) )
61	3, 53, 54, 5, 55, 56, 57, 13, 14, 15, 16, 58, 59, 60	cantnflem3 8120	. 2 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C e. ran ( A CNF B ) )
62	52, 61	rexlimddv 2963	1 |- ( ph -> C e. ran ( A CNF B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821   \ cdif 3478   C_ wss 3481   (/)c0 3790   ifcif 3944   <.cop 4038   U.cuni 4250   |^|cint 4287   {copab 4509   |-> cmpt 4510   Oncon0 4883   dom cdm 5004   ran crn 5005   iotacio 5554   Fn wfn 5588   -->wf 5589   ` cfv 5593  (class class class)co 6294   1stc1st 6792   2ndc2nd 6793   1oc1o 7133   2oc2o 7134   +o coa 7137   .o comu 7138   ^o coe 7139   CNF ccnf 8088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-seqom 7123  df-1o 7140  df-2o 7141  df-oadd 7144  df-omul 7145  df-oexp 7146  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-oi 7945  df-cnf 8089

This theorem is referenced by:  cantnf  8122