Theorem fidcenumlemrks 7236

Description: Lemma for fidcenum 7239. Induction step for fidcenumlemrk 7237. (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 3390	. . . . 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 4497	. . . . . . 7 |- suc J = ( J u. { J } )
5	4	imaeq2i 5104	. . . . . 6 |- ( F " suc J ) = ( F " ( J u. { J } ) )
6		imaundi 5180	. . . . . 6 |- ( F " ( J u. { J } ) ) = ( ( F " J ) u. ( F " { J } ) )
7	5, 6	eqtri 2255	. . . . 5 |- ( F " suc J ) = ( ( F " J ) u. ( F " { J } ) )
8	7	eleq2i 2301	. . . 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 741	. 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 3709	. . . . . . . . . 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 5597	. . . . . . . . . . . 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 4542	. . . . . . . . . . . . 13 |- ( J e. om -> J e. suc J )
21	19, 20	syl 14	. . . . . . . . . . . 12 |- ( ph -> J e. suc J )
22	18, 21	sseldd 3243	. . . . . . . . . . 11 |- ( ph -> J e. N )
23		fnsnfv 5741	. . . . . . . . . . 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 2304	. . . . . . . . 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 3391	. . . . . 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 741	. . 3 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ X = ( F ` J ) ) -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )
33		simplr 529	. . . . . . 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 679	. . . . . . 7 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> -. X e. ( F " { J } ) )
37		ioran 760	. . . . . . 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 3364	. . . . . 6 |- ( X e. ( ( F " J ) u. ( F " { J } ) ) <-> ( X e. ( F " J ) \/ X e. ( F " { J } ) ) )
40	38, 39	sylnibr 684	. . . . 5 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> -. X e. ( ( F " J ) u. ( F " { J } ) ) )
41	40, 8	sylnibr 684	. . . 4 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> -. X e. ( F " suc J ) )
42	41	olcd 742	. . 3 |- ( ( ( ph /\ -. X e. ( F " J ) ) /\ -. X = ( F ` J ) ) -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )
43		fof 5595	. . . . . . . 8 |- ( F : N -onto-> A -> F : N --> A )
44	15, 43	syl 14	. . . . . . 7 |- ( ph -> F : N --> A )
45	44, 22	ffvelcdmd 5818	. . . . . 6 |- ( ph -> ( F ` J ) e. A )
46		fidcenumlemr.dc	. . . . . 6 |- ( ph -> A. x e. A A. y e. A DECID x = y )
47		eqeq1 2241	. . . . . . . 8 |- ( x = X -> ( x = y <-> X = y ) )
48	47	dcbid 846	. . . . . . 7 |- ( x = X -> (DECID x = y <-> DECID X = y ) )
49		eqeq2 2244	. . . . . . . 8 |- ( y = ( F ` J ) -> ( X = y <-> X = ( F ` J ) ) )
50	49	dcbid 846	. . . . . . 7 |- ( y = ( F ` J ) -> (DECID X = y <-> DECID X = ( F ` J ) ) )
51	48, 50	rspc2va 2938	. . . . . 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 1273	. . . . 5 |- ( ph -> DECID X = ( F ` J ) )
53		exmiddc 844	. . . . 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 806	. 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 806	1 |- ( ph -> ( X e. ( F " suc J ) \/ -. X e. ( F " suc J ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   u. cun 3212   C_ wss 3214   {csn 3694   suc csuc 4491   omcom 4717   "cima 4757   Fn wfn 5352   -->wf 5353   -onto->wfo 5355   ` cfv 5357

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365

This theorem is referenced by:  fidcenumlemrk  7237