Theorem cantnflem4 7899

Description: Lemma for cantnf 7900. Complete the induction step of cantnflem3 7898. (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 7897	. . . . . . . . . . . 12 |- ( ph -> ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) )
9		eqid 2442	. . . . . . . . . . . . . 14 |- X = X
10		eqid 2442	. . . . . . . . . . . . . 14 |- Y = Y
11		eqid 2442	. . . . . . . . . . . . . 14 |- Z = Z
12	9, 10, 11	3pm3.2i 1166	. . . . . . . . . . . . 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 7040	. . . . . . . . . . . . 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 1000	. . . . . . . . 9 |- ( ph -> X e. On )
22		oecl 6976	. . . . . . . . 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 1001	. . . . . . . . . 10 |- ( ph -> Y e. ( A \ 1o ) )
25	24	eldifad 3339	. . . . . . . . 9 |- ( ph -> Y e. A )
26		onelon 4743	. . . . . . . . 9 |- ( ( A e. On /\ Y e. A ) -> Y e. On )
27	2, 25, 26	syl2anc 661	. . . . . . . 8 |- ( ph -> Y e. On )
28		omcl 6975	. . . . . . . 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 1002	. . . . . . . 8 |- ( ph -> Z e. ( A ^o X ) )
31		onelon 4743	. . . . . . . 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 6990	. . . . . . 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 6939	. . . . . . . . . . 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 4773	. . . . . . . . . 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 7011	. . . . . . . 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 1217	. . . . . . 7 |- ( ph -> ( A ^o X ) C_ ( ( A ^o X ) .o Y ) )
43	42, 30	sseldd 3356	. . . . . 6 |- ( ph -> Z e. ( ( A ^o X ) .o Y ) )
44	34, 43	sseldd 3356	. . . . 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 2518	. . . 4 |- ( ph -> Z e. C )
47	1, 46	sseldd 3356	. . 3 |- ( ph -> Z e. ran ( A CNF B ) )
48	3, 2, 4	cantnff 7881	. . . 4 |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )
49		ffn 5558	. . . 4 |- ( ( A CNF B ) : S --> ( A ^o B ) -> ( A CNF B ) Fn S )
50		fvelrnb 5738	. . . 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 2442	. . 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 7898	. 2 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C e. ran ( A CNF B ) )
62	52, 61	rexlimddv 2844	1 |- ( ph -> C e. ran ( A CNF B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2605   A.wral 2714   E.wrex 2715   {crab 2718   \ cdif 3324   C_ wss 3327   (/)c0 3636   ifcif 3790   <.cop 3882   U.cuni 4090   |^|cint 4127   {copab 4348   e. cmpt 4349   Oncon0 4718   dom cdm 4839   ran crn 4840   iotacio 5378   Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   1oc1o 6912   2oc2o 6913   +o coa 6916   .o comu 6917   ^o coe 6918   CNF ccnf 7866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6831  df-rdg 6865  df-seqom 6902  df-1o 6919  df-2o 6920  df-oadd 6923  df-omul 6924  df-oexp 6925  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-fsupp 7620  df-oi 7723  df-cnf 7867

This theorem is referenced by:  cantnf  7900