Theorem cantnflem4 8589

Description: Lemma for cantnf 8590. Complete the induction step of cantnflem3 8588. (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 8587	. . . . . . . . . . . 12 |- ( ph -> ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) )
9		eqid 2622	. . . . . . . . . . . . . 14 |- X = X
10		eqid 2622	. . . . . . . . . . . . . 14 |- Y = Y
11		eqid 2622	. . . . . . . . . . . . . 14 |- Z = Z
12	9, 10, 11	3pm3.2i 1239	. . . . . . . . . . . . 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 7682	. . . . . . . . . . . . 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 248	. . . . . . . . . . . 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 475	. . . . . . . . . 10 |- ( ph -> ( X e. On /\ Y e. ( A \ 1o ) /\ Z e. ( A ^o X ) ) )
21	20	simp1d 1073	. . . . . . . . 9 |- ( ph -> X e. On )
22		oecl 7617	. . . . . . . . 9 |- ( ( A e. On /\ X e. On ) -> ( A ^o X ) e. On )
23	2, 21, 22	syl2anc 693	. . . . . . . 8 |- ( ph -> ( A ^o X ) e. On )
24	20	simp2d 1074	. . . . . . . . . 10 |- ( ph -> Y e. ( A \ 1o ) )
25	24	eldifad 3586	. . . . . . . . 9 |- ( ph -> Y e. A )
26		onelon 5748	. . . . . . . . 9 |- ( ( A e. On /\ Y e. A ) -> Y e. On )
27	2, 25, 26	syl2anc 693	. . . . . . . 8 |- ( ph -> Y e. On )
28		omcl 7616	. . . . . . . 8 |- ( ( ( A ^o X ) e. On /\ Y e. On ) -> ( ( A ^o X ) .o Y ) e. On )
29	23, 27, 28	syl2anc 693	. . . . . . 7 |- ( ph -> ( ( A ^o X ) .o Y ) e. On )
30	20	simp3d 1075	. . . . . . . 8 |- ( ph -> Z e. ( A ^o X ) )
31		onelon 5748	. . . . . . . 8 |- ( ( ( A ^o X ) e. On /\ Z e. ( A ^o X ) ) -> Z e. On )
32	23, 30, 31	syl2anc 693	. . . . . . 7 |- ( ph -> Z e. On )
33		oaword1 7632	. . . . . . 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 693	. . . . . 6 |- ( ph -> ( ( A ^o X ) .o Y ) C_ ( ( ( A ^o X ) .o Y ) +o Z ) )
35		dif1o 7580	. . . . . . . . . . 11 |- ( Y e. ( A \ 1o ) <-> ( Y e. A /\ Y =/= (/) ) )
36	35	simprbi 480	. . . . . . . . . 10 |- ( Y e. ( A \ 1o ) -> Y =/= (/) )
37	24, 36	syl 17	. . . . . . . . 9 |- ( ph -> Y =/= (/) )
38		on0eln0 5780	. . . . . . . . . 10 |- ( Y e. On -> ( (/) e. Y <-> Y =/= (/) ) )
39	27, 38	syl 17	. . . . . . . . 9 |- ( ph -> ( (/) e. Y <-> Y =/= (/) ) )
40	37, 39	mpbird 247	. . . . . . . 8 |- ( ph -> (/) e. Y )
41		omword1 7653	. . . . . . . 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 1325	. . . . . . 7 |- ( ph -> ( A ^o X ) C_ ( ( A ^o X ) .o Y ) )
43	42, 30	sseldd 3604	. . . . . 6 |- ( ph -> Z e. ( ( A ^o X ) .o Y ) )
44	34, 43	sseldd 3604	. . . . 5 |- ( ph -> Z e. ( ( ( A ^o X ) .o Y ) +o Z ) )
45	19	simprd 479	. . . . 5 |- ( ph -> ( ( ( A ^o X ) .o Y ) +o Z ) = C )
46	44, 45	eleqtrd 2703	. . . 4 |- ( ph -> Z e. C )
47	1, 46	sseldd 3604	. . 3 |- ( ph -> Z e. ran ( A CNF B ) )
48	3, 2, 4	cantnff 8571	. . . 4 |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )
49		ffn 6045	. . . 4 |- ( ( A CNF B ) : S --> ( A ^o B ) -> ( A CNF B ) Fn S )
50		fvelrnb 6243	. . . 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 222	. 2 |- ( ph -> E. g e. S ( ( A CNF B ) ` g ) = Z )
53	2	adantr 481	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> A e. On )
54	4	adantr 481	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> B e. On )
55	6	adantr 481	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C e. ( A ^o B ) )
56	1	adantr 481	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C C_ ran ( A CNF B ) )
57	7	adantr 481	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> (/) e. C )
58		simprl 794	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> g e. S )
59		simprr 796	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> ( ( A CNF B ) ` g ) = Z )
60		eqid 2622	. . 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 8588	. 2 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C e. ran ( A CNF B ) )
62	52, 61	rexlimddv 3035	1 |- ( ph -> C e. ran ( A CNF B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   (/)c0 3915   ifcif 4086   <.cop 4183   U.cuni 4436   |^|cint 4475   {copab 4712   |-> cmpt 4729   dom cdm 5114   ran crn 5115   Oncon0 5723   iotacio 5849   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   1oc1o 7553   2oc2o 7554   +o coa 7557   .o comu 7558   ^o coe 7559   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-int 4476  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-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-seqom 7543  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566  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:  cantnf  8590