Theorem cantnflem4 8114

Description: Lemma for cantnf 8115. Complete the induction step of cantnflem3 8113. (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 8112	. . . . . . . . . . . 12 |- ( ph -> ( A e. ( On \ 2o ) /\ C e. ( On \ 1o ) ) )
9		eqid 2443	. . . . . . . . . . . . . 14 |- X = X
10		eqid 2443	. . . . . . . . . . . . . 14 |- Y = Y
11		eqid 2443	. . . . . . . . . . . . . 14 |- Z = Z
12	9, 10, 11	3pm3.2i 1175	. . . . . . . . . . . . 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 7253	. . . . . . . . . . . . 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 1009	. . . . . . . . 9 |- ( ph -> X e. On )
22		oecl 7189	. . . . . . . . 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 1010	. . . . . . . . . 10 |- ( ph -> Y e. ( A \ 1o ) )
25	24	eldifad 3473	. . . . . . . . 9 |- ( ph -> Y e. A )
26		onelon 4893	. . . . . . . . 9 |- ( ( A e. On /\ Y e. A ) -> Y e. On )
27	2, 25, 26	syl2anc 661	. . . . . . . 8 |- ( ph -> Y e. On )
28		omcl 7188	. . . . . . . 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 1011	. . . . . . . 8 |- ( ph -> Z e. ( A ^o X ) )
31		onelon 4893	. . . . . . . 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 7203	. . . . . . 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 7152	. . . . . . . . . . 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 4923	. . . . . . . . . 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 7224	. . . . . . . 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 1228	. . . . . . 7 |- ( ph -> ( A ^o X ) C_ ( ( A ^o X ) .o Y ) )
43	42, 30	sseldd 3490	. . . . . 6 |- ( ph -> Z e. ( ( A ^o X ) .o Y ) )
44	34, 43	sseldd 3490	. . . . 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 2533	. . . 4 |- ( ph -> Z e. C )
47	1, 46	sseldd 3490	. . 3 |- ( ph -> Z e. ran ( A CNF B ) )
48	3, 2, 4	cantnff 8096	. . . 4 |- ( ph -> ( A CNF B ) : S --> ( A ^o B ) )
49		ffn 5721	. . . 4 |- ( ( A CNF B ) : S --> ( A ^o B ) -> ( A CNF B ) Fn S )
50		fvelrnb 5905	. . . 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 756	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> g e. S )
59		simprr 757	. . 3 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> ( ( A CNF B ) ` g ) = Z )
60		eqid 2443	. . 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 8113	. 2 |- ( ( ph /\ ( g e. S /\ ( ( A CNF B ) ` g ) = Z ) ) -> C e. ran ( A CNF B ) )
62	52, 61	rexlimddv 2939	1 |- ( ph -> C e. ran ( A CNF B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   e. wcel 1804   =/= wne 2638   A.wral 2793   E.wrex 2794   {crab 2797   \ cdif 3458   C_ wss 3461   (/)c0 3770   ifcif 3926   <.cop 4020   U.cuni 4234   |^|cint 4271   {copab 4494   |-> cmpt 4495   Oncon0 4868   dom cdm 4989   ran crn 4990   iotacio 5539   Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784   1oc1o 7125   2oc2o 7126   +o coa 7129   .o comu 7130   ^o coe 7131   CNF ccnf 8081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-seqom 7115  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-oexp 7138  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-cnf 8082

This theorem is referenced by:  cantnf  8115