Theorem fidcenumlemrks 7088

Description: Lemma for fidcenum 7091. Induction step for fidcenumlemrk 7089. (Contributed by Jim Kingdon, 20-Oct-2022.)

Hypotheses

Ref	Expression
fidcenumlemr.dc	|- ( ph -> A. x e. A A. y e. A DECID x = y )
fidcenumlemr.f	|- ( ph -> F : N -onto-> A )
fidcenumlemrks.j	|- ( ph -> J e. om )
fidcenumlemrks.jn	|- ( ph -> suc J C_ N )
fidcenumlemrks.h	|- ( ph -> ( X e. ( F " J ) \/ -. X e. ( F " J ) ) )
fidcenumlemrks.x	|- ( ph -> X e. A )

Assertion

Ref	Expression
fidcenumlemrks	|- ( ph -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )

Distinct variable groups:   x, A, y   y, F   y, J   x, X, y

Allowed substitution hints:   ph( x, y)   F( x)   J( x)   N( x, y)

Proof of Theorem fidcenumlemrks

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( ( ph /\ X e. ( F " J ) ) -> X e. ( F " J ) )
2		elun1 3351	. . . . 5 |- ( X e. ( F " J ) -> X e. ( ( F " J ) u. ( F " { J } ) ) )
3	1, 2	syl 14	. . . 4 |- ( ( ph /\ X e. ( F " J ) ) -> X e. ( ( F " J ) u. ( F " { J } ) ) )
4		df-suc 4439	. . . . . . 7 |- suc J = ( J u. { J } )
5	4	imaeq2i 5042	. . . . . 6 |- ( F " suc J ) = ( F " ( J u. { J } ) )
6		imaundi 5117	. . . . . 6 |- ( F " ( J u. { J } ) ) = ( ( F " J ) u. ( F " { J } ) )
7	5, 6	eqtri 2230	. . . . 5 |- ( F " suc J ) = ( ( F " J ) u. ( F " { J } ) )
8	7	eleq2i 2276	. . . 4 |- ( X e. ( F " suc J ) <-> X e. ( ( F " J ) u. ( F " { J } ) ) )
9	3, 8	sylibr 134	. . 3 |- ( ( ph /\ X e. ( F " J ) ) -> X e. ( F " suc J ) )
10	9	orcd 737	. 2 |- ( ( ph /\ X e. ( F " J ) ) -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )
11		simpr 110	. . . . . . 7 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ X = ( F ` J ) ) -> X = ( F ` J ) )
12		fidcenumlemrks.x	. . . . . . . . . 10 |- ( ph -> X e. A )
13		elsng 3661	. . . . . . . . . 10 |- ( X e. A -> ( X e. { ( F ` J ) } <-> X = ( F ` J ) ) )
14	12, 13	syl 14	. . . . . . . . 9 |- ( ph -> ( X e. { ( F ` J ) } <-> X = ( F ` J ) ) )
15		fidcenumlemr.f	. . . . . . . . . . . 12 |- ( ph -> F : N -onto-> A )
16		fofn 5526	. . . . . . . . . . . 12 |- ( F : N -onto-> A -> F Fn N )
17	15, 16	syl 14	. . . . . . . . . . 11 |- ( ph -> F Fn N )
18		fidcenumlemrks.jn	. . . . . . . . . . . 12 |- ( ph -> suc J C_ N )
19		fidcenumlemrks.j	. . . . . . . . . . . . 13 |- ( ph -> J e. om )
20		sucidg 4484	. . . . . . . . . . . . 13 |- ( J e. om -> J e. suc J )
21	19, 20	syl 14	. . . . . . . . . . . 12 |- ( ph -> J e. suc J )
22	18, 21	sseldd 3205	. . . . . . . . . . 11 |- ( ph -> J e. N )
23		fnsnfv 5666	. . . . . . . . . . 11 |- ( ( F Fn N /\ J e. N ) -> { ( F ` J ) } = ( F " { J } ) )
24	17, 22, 23	syl2anc 411	. . . . . . . . . 10 |- ( ph -> { ( F ` J ) } = ( F " { J } ) )
25	24	eleq2d 2279	. . . . . . . . 9 |- ( ph -> ( X e. { ( F ` J ) } <-> X e. ( F " { J } ) ) )
26	14, 25	bitr3d 190	. . . . . . . 8 |- ( ph -> ( X = ( F ` J ) <-> X e. ( F " { J } ) ) )
27	26	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ X = ( F ` J ) ) -> ( X = ( F ` J ) <-> X e. ( F " { J } ) ) )
28	11, 27	mpbid 147	. . . . . 6 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ X = ( F ` J ) ) -> X e. ( F " { J } ) )
29		elun2 3352	. . . . . 6 |- ( X e. ( F " { J } ) -> X e. ( ( F " J ) u. ( F " { J } ) ) )
30	28, 29	syl 14	. . . . 5 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ X = ( F ` J ) ) -> X e. ( ( F " J ) u. ( F " { J } ) ) )
31	30, 8	sylibr 134	. . . 4 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ X = ( F ` J ) ) -> X e. ( F " suc J ) )
32	31	orcd 737	. . 3 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ X = ( F ` J ) ) -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )
33		simplr 528	. . . . . . 7 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> -. X e. ( F " J ) )
34		simpr 110	. . . . . . . 8 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> -. X = ( F ` J ) )
35	26	ad2antrr 488	. . . . . . . 8 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> ( X = ( F ` J ) <-> X e. ( F " { J } ) ) )
36	34, 35	mtbid 676	. . . . . . 7 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> -. X e. ( F " { J } ) )
37		ioran 756	. . . . . . 7 |- ( -. ( X e. ( F " J ) \/ X e. ( F " { J } ) ) <-> ( -. X e. ( F " J ) /\ -. X e. ( F " { J } ) ) )
38	33, 36, 37	sylanbrc 417	. . . . . 6 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> -. ( X e. ( F " J ) \/ X e. ( F " { J } ) ) )
39		elun 3325	. . . . . 6 |- ( X e. ( ( F " J ) u. ( F " { J } ) ) <-> ( X e. ( F " J ) \/ X e. ( F " { J } ) ) )
40	38, 39	sylnibr 681	. . . . 5 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> -. X e. ( ( F " J ) u. ( F " { J } ) ) )
41	40, 8	sylnibr 681	. . . 4 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> -. X e. ( F " suc J ) )
42	41	olcd 738	. . 3 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )
43		fof 5524	. . . . . . . 8 |- ( F : N -onto-> A -> F : N --> A )
44	15, 43	syl 14	. . . . . . 7 |- ( ph -> F : N --> A )
45	44, 22	ffvelcdmd 5744	. . . . . 6 |- ( ph -> ( F ` J ) e. A )
46		fidcenumlemr.dc	. . . . . 6 |- ( ph -> A. x e. A A. y e. A DECID x = y )
47		eqeq1 2216	. . . . . . . 8 |- ( x = X -> ( x = y <-> X = y ) )
48	47	dcbid 842	. . . . . . 7 |- ( x = X -> (DECID x = y <-> DECID X = y ) )
49		eqeq2 2219	. . . . . . . 8 |- ( y = ( F ` J ) -> ( X = y <-> X = ( F ` J ) ) )
50	49	dcbid 842	. . . . . . 7 |- ( y = ( F ` J ) -> (DECID X = y <-> DECID X = ( F ` J ) ) )
51	48, 50	rspc2va 2901	. . . . . 6 |- ( ( ( X e. A /\ ( F ` J ) e. A ) /\ A. x e. A A. y e. A DECID x = y ) -> DECID X = ( F ` J ) )
52	12, 45, 46, 51	syl21anc 1251	. . . . 5 |- ( ph -> DECID X = ( F ` J ) )
53		exmiddc 840	. . . . 5 |- (DECID X = ( F ` J ) -> ( X = ( F ` J ) \/ -. X = ( F ` J ) ) )
54	52, 53	syl 14	. . . 4 |- ( ph -> ( X = ( F ` J ) \/ -. X = ( F ` J ) ) )
55	54	adantr 276	. . 3 |- ( ( ph /\ -. X e. ( F " J ) ) -> ( X = ( F ` J ) \/ -. X = ( F ` J ) ) )
56	32, 42, 55	mpjaodan 802	. 2 |- ( ( ph /\ -. X e. ( F " J ) ) -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )
57		fidcenumlemrks.h	. 2 |- ( ph -> ( X e. ( F " J ) \/ -. X e. ( F " J ) ) )
58	10, 56, 57	mpjaodan 802	1 |- ( ph -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 712  DECID wdc 838   = wceq 1375   e. wcel 2180   A.wral 2488   u. cun 3175   C_ wss 3177   {csn 3646   suc csuc 4433   omcom 4659   "cima 4699   Fn wfn 5289   -->wf 5290   -onto->wfo 5292   ` cfv 5294

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-id 4361  df-suc 4439  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-fo 5300  df-fv 5302

This theorem is referenced by:  fidcenumlemrk  7089