Theorem ctiunctlemudc 13306

Description: Lemma for ctiunct 13309. (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 2301	. . . . . . . . 9 |- ( n = ( 1st ` ( J ` m ) ) -> ( n e. S <-> ( 1st ` ( J ` m ) ) e. S ) )
2	1	dcbid 850	. . . . . . . 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 276	. . . . . . . 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 276	. . . . . . . . . . 11 |- ( ( ph /\ m e. om ) -> J : om -1-1-onto-> ( om X. om ) )
7		f1of 5634	. . . . . . . . . . 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 110	. . . . . . . . . 10 |- ( ( ph /\ m e. om ) -> m e. om )
10	8, 9	ffvelcdmd 5835	. . . . . . . . 9 |- ( ( ph /\ m e. om ) -> ( J ` m ) e. ( om X. om ) )
11		xp1st 6389	. . . . . . . . 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 2934	. . . . . . 7 |- ( ( ph /\ m e. om ) -> DECID ( 1st ` ( J ` m ) ) e. S )
14	13	adantr 276	. . . . . 6 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> DECID ( 1st ` ( J ` m ) ) e. S )
15		eleq1 2301	. . . . . . . 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 850	. . . . . . 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 5610	. . . . . . . . . . 11 |- ( F : S -onto-> A -> F : S --> A )
19	17, 18	syl 14	. . . . . . . . . 10 |- ( ph -> F : S --> A )
20	19	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> F : S --> A )
21		simpr 110	. . . . . . . . 9 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> ( 1st ` ( J ` m ) ) e. S )
22	20, 21	ffvelcdmd 5835	. . . . . . . 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 2623	. . . . . . . . 9 |- ( ph -> A. x e. A A. n e. om DECID n e. T )
25	24	ad2antrr 492	. . . . . . . 8 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> A. x e. A A. n e. om DECID n e. T )
26		nfcv 2392	. . . . . . . . . 10 |- F/_ x om
27		nfcsb1v 3180	. . . . . . . . . . . 12 |- F/_ x [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
28	27	nfcri 2386	. . . . . . . . . . 11 |- F/ x n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
29	28	nfdc 1711	. . . . . . . . . 10 |- F/ xDECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
30	26, 29	nfralya 2590	. . . . . . . . 9 |- F/ x A. n e. om DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T
31		csbeq1a 3156	. . . . . . . . . . . 12 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> T = [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
32	31	eleq2d 2308	. . . . . . . . . . 11 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> ( n e. T <-> n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
33	32	dcbid 850	. . . . . . . . . 10 |- ( x = ( F ` ( 1st ` ( J ` m ) ) ) -> (DECID n e. T <-> DECID n e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
34	33	ralbidv 2550	. . . . . . . . 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 2923	. . . . . . . 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 276	. . . . . . . 8 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> ( J ` m ) e. ( om X. om ) )
38		xp2nd 6390	. . . . . . . 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 2934	. . . . . 6 |- ( ( ( ph /\ m e. om ) /\ ( 1st ` ( J ` m ) ) e. S ) -> DECID ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
41		dcan2 947	. . . . . 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 110	. . . . . . . 8 |- ( ( ( ph /\ m e. om ) /\ -. ( 1st ` ( J ` m ) ) e. S ) -> -. ( 1st ` ( J ` m ) ) e. S )
44	43	intnanrd 944	. . . . . . 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 746	. . . . . 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 847	. . . . . 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 134	. . . . 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 848	. . . . . 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 810	. . . 4 |- ( ( ph /\ m e. om ) -> DECID ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) )
51		2fveq3 5695	. . . . . . . . 9 |- ( z = m -> ( 1st ` ( J ` z ) ) = ( 1st ` ( J ` m ) ) )
52	51	eleq1d 2307	. . . . . . . 8 |- ( z = m -> ( ( 1st ` ( J ` z ) ) e. S <-> ( 1st ` ( J ` m ) ) e. S ) )
53		2fveq3 5695	. . . . . . . . 9 |- ( z = m -> ( 2nd ` ( J ` z ) ) = ( 2nd ` ( J ` m ) ) )
54	51	fveq2d 5694	. . . . . . . . . 10 |- ( z = m -> ( F ` ( 1st ` ( J ` z ) ) ) = ( F ` ( 1st ` ( J ` m ) ) ) )
55	54	csbeq1d 3154	. . . . . . . . 9 |- ( z = m -> [_ ( F ` ( 1st ` ( J ` z ) ) ) / x ]_ T = [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T )
56	53, 55	eleq12d 2309	. . . . . . . 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 477	. . . . . . 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 2985	. . . . . 6 |- ( m e. U <-> ( m e. om /\ ( ( 1st ` ( J ` m ) ) e. S /\ ( 2nd ` ( J ` m ) ) e. [_ ( F ` ( 1st ` ( J ` m ) ) ) / x ]_ T ) ) )
60		ibar 301	. . . . . . 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 277	. . . . . 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 199	. . . . 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 850	. . . 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 167	. . 3 |- ( ( ph /\ m e. om ) -> DECID m e. U )
65	64	ralrimiva 2623	. 2 |- ( ph -> A. m e. om DECID m e. U )
66		eleq1 2301	. . . 4 |- ( m = n -> ( m e. U <-> n e. U ) )
67	66	dcbid 850	. . 3 |- ( m = n -> (DECID m e. U <-> DECID n e. U ) )
68	67	cbvralv 2786	. 2 |- ( A. m e. om DECID m e. U <-> A. n e. om DECID n e. U )
69	65, 68	sylib 122	1 |- ( ph -> A. n e. om DECID n e. U )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   [_csb 3147   C_ wss 3220   omcom 4732   X. cxp 4767   -->wf 5368   -onto->wfo 5370   -1-1-onto->wf1o 5371   ` cfv 5372   1stc1st 6362   2ndc2nd 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365

This theorem is referenced by:  ctiunct  13309