Theorem ctiunctlemudc 11986

Description: Lemma for ctiunct 11989. (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 2203	. . . . . . . . 9 |- ( n = ( 1st ` ( J ` m ) ) -> ( n e. S <-> ( 1st ` ( J ` m ) ) e. S ) )
2	1	dcbid 824	. . . . . . . 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 5375	. . . . . . . . . . 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 5564	. . . . . . . . 9 |- ( ( ph /\ m e. om ) -> ( J ` m ) e. ( om X. om ) )
11		xp1st 6071	. . . . . . . . 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 2798	. . . . . . 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 2203	. . . . . . . 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 824	. . . . . . 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 5353	. . . . . . . . . . 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 5564	. . . . . . . 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 2508	. . . . . . . . 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 2282	. . . . . . . . . 10 |- F/_ x om
27		nfcsb1v 3040	. . . . . . . . . . . 12 |- F/_ x [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
28	27	nfcri 2276	. . . . . . . . . . 11 |- F/ x n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
29	28	nfdc 1638	. . . . . . . . . 10 |- F/ xDECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
30	26, 29	nfralya 2476	. . . . . . . . 9 |- F/ x A. n e. om DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
31		csbeq1a 3016	. . . . . . . . . . . 12 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> T = [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
32	31	eleq2d 2210	. . . . . . . . . . 11 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> ( n e. T <-> n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
33	32	dcbid 824	. . . . . . . . . 10 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> (DECID n e. T <-> DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
34	33	ralbidv 2438	. . . . . . . . 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 2787	. . . . . . . 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 6072	. . . . . . . 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 2798	. . . . . 6 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> DECID ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
41		dcan 919	. . . . . 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 918	. . . . . . 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 821	. . . . . 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 822	. . . . . 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 5434	. . . . . . . . 9 |- ( z = m -> ( 1st ` ( J ` z ) ) = ( 1st ` ( J ` m ) ) )
52	51	eleq1d 2209	. . . . . . . 8 |- ( z = m -> ( ( 1st ` ( J ` z ) ) e. S <-> ( 1st ` ( J ` m ) ) e. S ) )
53		2fveq3 5434	. . . . . . . . 9 |- ( z = m -> ( 2nd ` ( J ` z ) ) = ( 2nd ` ( J ` m ) ) )
54	51	fveq2d 5433	. . . . . . . . . 10 |- ( z = m -> ( F ` ( 1st ` ( J ` z ) ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
55	54	csbeq1d 3014	. . . . . . . . 9 |- ( z = m -> [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T = [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
56	53, 55	eleq12d 2211	. . . . . . . 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 2847	. . . . . 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 824	. . . 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 2508	. 2 |- ( ph -> A. m e. om DECID m e. U )
66		eleq1 2203	. . . 4 |- ( m = n -> ( m e. U <-> n e. U ) )
67	66	dcbid 824	. . 3 |- ( m = n -> (DECID m e. U <-> DECID n e. U ) )
68	67	cbvralv 2657	. 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 820   = wceq 1332   e. wcel 1481   A.wral 2417   {crab 2421   [_csb 3007   C_ wss 3076   omcom 4512   X. cxp 4545   -->wf 5127   -onto->wfo 5129   -1-1-onto->wf1o 5130   ` cfv 5131   1stc1st 6044   2ndc2nd 6045

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-1st 6046  df-2nd 6047

This theorem is referenced by:  ctiunct  11989