Theorem cantnflem4 8197

Description: Lemma for cantnf 8198. Complete the induction step of cantnflem3 8196. (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 8195	. . . . . . . . . . . 12 |- ( ph -> ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) )
9		eqid 2451	. . . . . . . . . . . . . 14 |- X = X
10		eqid 2451	. . . . . . . . . . . . . 14 |- Y = Y
11		eqid 2451	. . . . . . . . . . . . . 14 |- Z = Z
12	9, 10, 11	3pm3.2i 1186	. . . . . . . . . . . . 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 7303	. . . . . . . . . . . . 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 237	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 461	. . . . . . . . . 10 |- ( ph -> ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) )
21	20	simp1d 1020	. . . . . . . . 9 |- ( ph -> X e. On )
22		oecl 7239	. . . . . . . . 9 |- ( ( A e. On /\ X e. On ) -> ( A ^o X ) e. On )
23	2, 21, 22	syl2anc 667	. . . . . . . 8 |- ( ph -> ( A ^o X ) e. On )
24	20	simp2d 1021	. . . . . . . . . 10 |- ( ph -> Y e. ( A \ 1o ) )
25	24	eldifad 3416	. . . . . . . . 9 |- ( ph -> Y e. A )
26		onelon 5448	. . . . . . . . 9 |- ( ( A e. On /\ Y e. A ) -> Y e. On )
27	2, 25, 26	syl2anc 667	. . . . . . . 8 |- ( ph -> Y e. On )
28		omcl 7238	. . . . . . . 8 |- ( ( ( A ^o X ) e. On /\ Y e. On ) -> ( ( A ^o X ) .o Y ) e. On )
29	23, 27, 28	syl2anc 667	. . . . . . 7 |- ( ph -> ( ( A ^o X ) .o Y ) e. On )
30	20	simp3d 1022	. . . . . . . 8 |- ( ph -> Z e. ( A ^o X ) )
31		onelon 5448	. . . . . . . 8 |- ( ( ( A ^o X ) e. On /\ Z e. ( A ^o X ) ) -> Z e. On )
32	23, 30, 31	syl2anc 667	. . . . . . 7 |- ( ph -> Z e. On )
33		oaword1 7253	. . . . . . 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 667	. . . . . 6 |- ( ph -> ( ( A ^o X ) .o Y ) C_ ( ( ( A ^o X ) .o Y ) +o Z ) )
35		dif1o 7202	. . . . . . . . . . 11 |- ( Y e. ( A \ 1o ) <-> ( Y e. A /\ Y =/= (/) ) )
36	35	simprbi 466	. . . . . . . . . 10 |- ( Y e. ( A \ 1o ) -> Y =/= (/) )
37	24, 36	syl 17	. . . . . . . . 9 |- ( ph -> Y =/= (/) )
38		on0eln0 5478	. . . . . . . . . 10 |- ( Y e. On -> ( (/) e. Y <-> Y =/= (/) ) )
39	27, 38	syl 17	. . . . . . . . 9 |- ( ph -> ( (/) e. Y <-> Y =/= (/) ) )
40	37, 39	mpbird 236	. . . . . . . 8 |- ( ph -> (/) e. Y )
41		omword1 7274	. . . . . . . 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 1267	. . . . . . 7 |- ( ph -> ( A ^o X ) C_ ( ( A ^o X ) .o Y ) )
43	42, 30	sseldd 3433	. . . . . 6 |- ( ph -> Z e. ( ( A ^o X ) .o Y ) )
44	34, 43	sseldd 3433	. . . . 5 |- ( ph -> Z e. ( ( ( A ^o X ) .o Y ) +o Z ) )
45	19	simprd 465	. . . . 5 |- ( ph -> ( ( ( A ^o X ) .o Y ) +o Z ) = C )
46	44, 45	eleqtrd 2531	. . . 4 |- ( ph -> Z e. C )
47	1, 46	sseldd 3433	. . 3 |- ( ph -> Z e. ran ( A CNF B ) )
48	3, 2, 4	cantnff 8179	. . . 4 |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )
49		ffn 5728	. . . 4 |- ( ( A CNF B ) : S --> ( A ^o B ) -> ( A CNF B ) Fn S )
50		fvelrnb 5912	. . . 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 18	. . 3 |- ( ph -> ( Z e. ran ( A CNF B ) <-> E. g e. S ( ( A CNF B ) ` g ) = Z ) )
52	47, 51	mpbid 214	. 2 |- ( ph -> E. g e. S ( ( A CNF B ) ` g ) = Z )
53	2	adantr 467	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> A e. On )
54	4	adantr 467	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> B e. On )
55	6	adantr 467	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C e. ( A ^o B ) )
56	1	adantr 467	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C C_ ran ( A CNF B ) )
57	7	adantr 467	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> (/) e. C )
58		simprl 764	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> g e. S )
59		simprr 766	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> ( ( A CNF B ) ` g ) = Z )
60		eqid 2451	. . 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 8196	. 2 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C e. ran ( A CNF B ) )
62	52, 61	rexlimddv 2883	1 |- ( ph -> C e. ran ( A CNF B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741   \ cdif 3401   C_ wss 3404   (/)c0 3731   ifcif 3881   <.cop 3974   U.cuni 4198   |^|cint 4234   {copab 4460   |-> cmpt 4461   dom cdm 4834   ran crn 4835   Oncon0 5423   iotacio 5544   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792   1oc1o 7175   2oc2o 7176   +o coa 7179   .o comu 7180   ^o coe 7181   CNF ccnf 8166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-seqom 7165  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-oexp 7188  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-oi 8025  df-cnf 8167

This theorem is referenced by:  cantnf  8198