Theorem ctiunctlemudc 12370

Description: Lemma for ctiunct 12373. (Contributed by Jim Kingdon, 28-Oct-2023.)

Hypotheses

Ref	Expression
ctiunct.som	|- ( ph -> S C_ om )
ctiunct.sdc	|- ( ph -> A. n e. om DECID n e. S )
ctiunct.f	|- ( ph -> F : S -onto-> A )
ctiunct.tom	|- ( ( ph /\ x e. A ) -> T C_ om )
ctiunct.tdc	|- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )
ctiunct.g	|- ( ( ph /\ x e. A ) -> G : T -onto-> B )
ctiunct.j	|- ( ph -> J : om -1-1-onto-> ( om X. om ) )
ctiunct.u	|- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }

Assertion

Ref	Expression
ctiunctlemudc	|- ( ph -> A. n e. om DECID n e. U )

Distinct variable groups:   x, A   n, F, x   z, F, x   n, J, x   z, J   S, n   z, S   T, n   z, T   U, n   ph, x

Allowed substitution hints:   ph( z, n)   A( z, n)   B( x, z, n)   S( x)   T( x)   U( x, z)   G( x, z, n)

Proof of Theorem ctiunctlemudc

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2229	. . . . . . . . 9 |- ( n = ( 1st ` ( J ` m ) ) -> ( n e. S <-> ( 1st ` ( J ` m ) ) e. S ) )
2	1	dcbid 828	. . . . . . . 8 |- ( n = ( 1st ` ( J ` m ) ) -> (DECID n e. S <-> DECID ( 1st ` ( J ` m ) ) e. S ) )
3		ctiunct.sdc	. . . . . . . . 9 |- ( ph -> A. n e. om DECID n e. S )
4	3	adantr 274	. . . . . . . 8 |- ( ( ph /\ m e. om ) -> A. n e. om DECID n e. S )
5		ctiunct.j	. . . . . . . . . . . 12 |- ( ph -> J : om -1-1-onto-> ( om X. om ) )
6	5	adantr 274	. . . . . . . . . . 11 |- ( ( ph /\ m e. om ) -> J : om -1-1-onto-> ( om X. om ) )
7		f1of 5432	. . . . . . . . . . 11 |- ( J : om -1-1-onto-> ( om X. om ) -> J : om --> ( om X. om ) )
8	6, 7	syl 14	. . . . . . . . . 10 |- ( ( ph /\ m e. om ) -> J : om --> ( om X. om ) )
9		simpr 109	. . . . . . . . . 10 |- ( ( ph /\ m e. om ) -> m e. om )
10	8, 9	ffvelrnd 5621	. . . . . . . . 9 |- ( ( ph /\ m e. om ) -> ( J ` m ) e. ( om X. om ) )
11		xp1st 6133	. . . . . . . . 9 |- ( ( J ` m ) e. ( om X. om ) -> ( 1st ` ( J ` m ) ) e. om )
12	10, 11	syl 14	. . . . . . . 8 |- ( ( ph /\ m e. om ) -> ( 1st ` ( J ` m ) ) e. om )
13	2, 4, 12	rspcdva 2835	. . . . . . 7 |- ( ( ph /\ m e. om ) -> DECID ( 1st ` ( J ` m ) ) e. S )
14	13	adantr 274	. . . . . 6 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> DECID ( 1st ` ( J ` m ) ) e. S )
15		eleq1 2229	. . . . . . . 8 |- ( n = ( 2nd ` ( J ` m ) ) -> ( n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T <-> ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
16	15	dcbid 828	. . . . . . 7 |- ( n = ( 2nd ` ( J ` m ) ) -> (DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T <-> DECID ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
17		ctiunct.f	. . . . . . . . . . 11 |- ( ph -> F : S -onto-> A )
18		fof 5410	. . . . . . . . . . 11 |- ( F : S -onto-> A -> F : S --> A )
19	17, 18	syl 14	. . . . . . . . . 10 |- ( ph -> F : S --> A )
20	19	ad2antrr 480	. . . . . . . . 9 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> F : S --> A )
21		simpr 109	. . . . . . . . 9 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> ( 1st ` ( J ` m ) ) e. S )
22	20, 21	ffvelrnd 5621	. . . . . . . 8 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> ( F ` ( 1st ` ( J ` m ) ) ) e. A )
23		ctiunct.tdc	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> A. n e. om DECID n e. T )
24	23	ralrimiva 2539	. . . . . . . . 9 |- ( ph -> A. x e. A A. n e. om DECID n e. T )
25	24	ad2antrr 480	. . . . . . . 8 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> A. x e. A A. n e. om DECID n e. T )
26		nfcv 2308	. . . . . . . . . 10 |- F/_ x om
27		nfcsb1v 3078	. . . . . . . . . . . 12 |- F/_ x [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
28	27	nfcri 2302	. . . . . . . . . . 11 |- F/ x n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
29	28	nfdc 1647	. . . . . . . . . 10 |- F/ xDECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
30	26, 29	nfralya 2506	. . . . . . . . 9 |- F/ x A. n e. om DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
31		csbeq1a 3054	. . . . . . . . . . . 12 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> T = [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
32	31	eleq2d 2236	. . . . . . . . . . 11 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> ( n e. T <-> n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
33	32	dcbid 828	. . . . . . . . . 10 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> (DECID n e. T <-> DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
34	33	ralbidv 2466	. . . . . . . . 9 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> ( A. n e. om DECID n e. T <-> A. n e. om DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
35	30, 34	rspc 2824	. . . . . . . 8 |- ( ( F ` ( 1st ` ( J ` m ) ) ) e. A -> ( A. x e. A A. n e. om DECID n e. T -> A. n e. om DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
36	22, 25, 35	sylc 62	. . . . . . 7 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> A. n e. om DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
37	10	adantr 274	. . . . . . . 8 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> ( J ` m ) e. ( om X. om ) )
38		xp2nd 6134	. . . . . . . 8 |- ( ( J ` m ) e. ( om X. om ) -> ( 2nd ` ( J ` m ) ) e. om )
39	37, 38	syl 14	. . . . . . 7 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> ( 2nd ` ( J ` m ) ) e. om )
40	16, 36, 39	rspcdva 2835	. . . . . 6 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> DECID ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
41		dcan2 924	. . . . . 6 |- (DECID ( 1st ` ( J ` m ) ) e. S -> (DECID ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T -> DECID ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) )
42	14, 40, 41	sylc 62	. . . . 5 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> DECID ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
43		simpr 109	. . . . . . . 8 |- ( ( ( ph /\ m e. om ) /\ -. ( 1st ` ( J ` m ) ) e. S ) -> -. ( 1st ` ( J ` m ) ) e. S )
44	43	intnanrd 922	. . . . . . 7 |- ( ( ( ph /\ m e. om ) /\ -. ( 1st ` ( J ` m ) ) e. S ) -> -. ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
45	44	olcd 724	. . . . . 6 |- ( ( ( ph /\ m e. om ) /\ -. ( 1st ` ( J ` m ) ) e. S ) -> ( ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) \/ -. ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) )
46		df-dc 825	. . . . . 6 |- (DECID ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) <-> ( ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) \/ -. ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) )
47	45, 46	sylibr 133	. . . . 5 |- ( ( ( ph /\ m e. om ) /\ -. ( 1st ` ( J ` m ) ) e. S ) -> DECID ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
48		exmiddc 826	. . . . . 6 |- (DECID ( 1st ` ( J ` m ) ) e. S -> ( ( 1st ` ( J ` m ) ) e. S \/ -. ( 1st ` ( J ` m ) ) e. S ) )
49	13, 48	syl 14	. . . . 5 |- ( ( ph /\ m e. om ) -> ( ( 1st ` ( J ` m ) ) e. S \/ -. ( 1st ` ( J ` m ) ) e. S ) )
50	42, 47, 49	mpjaodan 788	. . . 4 |- ( ( ph /\ m e. om ) -> DECID ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
51		2fveq3 5491	. . . . . . . . 9 |- ( z = m -> ( 1st ` ( J ` z ) ) = ( 1st ` ( J ` m ) ) )
52	51	eleq1d 2235	. . . . . . . 8 |- ( z = m -> ( ( 1st ` ( J ` z ) ) e. S <-> ( 1st ` ( J ` m ) ) e. S ) )
53		2fveq3 5491	. . . . . . . . 9 |- ( z = m -> ( 2nd ` ( J ` z ) ) = ( 2nd ` ( J ` m ) ) )
54	51	fveq2d 5490	. . . . . . . . . 10 |- ( z = m -> ( F ` ( 1st ` ( J ` z ) ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
55	54	csbeq1d 3052	. . . . . . . . 9 |- ( z = m -> [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T = [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
56	53, 55	eleq12d 2237	. . . . . . . 8 |- ( z = m -> ( ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T <-> ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
57	52, 56	anbi12d 465	. . . . . . 7 |- ( z = m -> ( ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) <-> ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) )
58		ctiunct.u	. . . . . . 7 |- U = { z e. om | ( ( 1st ` ( J ` z ) ) e. S /\ ( 2nd ` ( J ` z ) ) e. [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T ) }
59	57, 58	elrab2 2885	. . . . . 6 |- ( m e. U <-> ( m e. om /\ ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) )
60		ibar 299	. . . . . . 7 |- ( m e. om -> ( ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) <-> ( m e. om /\ ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) ) )
61	60	adantl 275	. . . . . 6 |- ( ( ph /\ m e. om ) -> ( ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) <-> ( m e. om /\ ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) ) )
62	59, 61	bitr4id 198	. . . . 5 |- ( ( ph /\ m e. om ) -> ( m e. U <-> ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) )
63	62	dcbid 828	. . . 4 |- ( ( ph /\ m e. om ) -> (DECID m e. U <-> DECID ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) )
64	50, 63	mpbird 166	. . 3 |- ( ( ph /\ m e. om ) -> DECID m e. U )
65	64	ralrimiva 2539	. 2 |- ( ph -> A. m e. om DECID m e. U )
66		eleq1 2229	. . . 4 |- ( m = n -> ( m e. U <-> n e. U ) )
67	66	dcbid 828	. . 3 |- ( m = n -> (DECID m e. U <-> DECID n e. U ) )
68	67	cbvralv 2692	. 2 |- ( A. m e. om DECID m e. U <-> A. n e. om DECID n e. U )
69	65, 68	sylib 121	1 |- ( ph -> A. n e. om DECID n e. U )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   = wceq 1343   e. wcel 2136   A.wral 2444   {crab 2448   [_csb 3045   C_ wss 3116   omcom 4567   X. cxp 4602   -->wf 5184   -onto->wfo 5186   -1-1-onto->wf1o 5187   ` cfv 5188   1stc1st 6106   2ndc2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109

This theorem is referenced by:  ctiunct  12373