Theorem ennnfonelemr 12427

Description: Lemma for ennnfone 12429. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.)

Hypotheses

Ref	Expression
ennnfonelemr.dceq	|- ( ph -> A. x e. A A. y e. A DECID x = y )
ennnfonelemr.f	|- ( ph -> F : NN0 -onto-> A )
ennnfonelemr.n	|- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )

Assertion

Ref	Expression
ennnfonelemr	|- ( ph -> A ~~ NN )

Distinct variable groups:   y, A, x   n, F, j, k

Allowed substitution hints:   ph( x, y, j, k, n)   A( j, k, n)   F( x, y)

Proof of Theorem ennnfonelemr

Dummy variables a b d e f c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemr.dceq	. . 3 |- ( ph -> A. x e. A A. y e. A DECID x = y )
2		equequ1 1712	. . . . 5 |- ( x = a -> ( x = y <-> a = y ) )
3	2	dcbid 838	. . . 4 |- ( x = a -> (DECID x = y <-> DECID a = y ) )
4		equequ2 1713	. . . . 5 |- ( y = b -> ( a = y <-> a = b ) )
5	4	dcbid 838	. . . 4 |- ( y = b -> (DECID a = y <-> DECID a = b ) )
6	3, 5	cbvral2v 2718	. . 3 |- ( A. x e. A A. y e. A DECID x = y <-> A. a e. A A. b e. A DECID a = b )
7	1, 6	sylib 122	. 2 |- ( ph -> A. a e. A A. b e. A DECID a = b )
8		ennnfonelemr.f	. 2 |- ( ph -> F : NN0 -onto-> A )
9		ennnfonelemr.n	. . 3 |- ( ph -> A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) )
10		fveq2 5517	. . . . . . . . 9 |- ( j = f -> ( F ` j ) = ( F ` f ) )
11	10	neeq2d 2366	. . . . . . . 8 |- ( j = f -> ( ( F ` k ) =/= ( F ` j ) <-> ( F ` k ) =/= ( F ` f ) ) )
12	11	cbvralv 2705	. . . . . . 7 |- ( A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) )
13	12	rexbii 2484	. . . . . 6 |- ( E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> E. k e. NN0 A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) )
14		fveq2 5517	. . . . . . . . 9 |- ( k = e -> ( F ` k ) = ( F ` e ) )
15	14	neeq1d 2365	. . . . . . . 8 |- ( k = e -> ( ( F ` k ) =/= ( F ` f ) <-> ( F ` e ) =/= ( F ` f ) ) )
16	15	ralbidv 2477	. . . . . . 7 |- ( k = e -> ( A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) <-> A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) ) )
17	16	cbvrexv 2706	. . . . . 6 |- ( E. k e. NN0 A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) <-> E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) )
18	13, 17	bitri 184	. . . . 5 |- ( E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) )
19	18	ralbii 2483	. . . 4 |- ( A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> A. n e. NN0 E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) )
20		oveq2 5886	. . . . . . 7 |- ( n = d -> ( 0 ... n ) = ( 0 ... d ) )
21	20	raleqdv 2679	. . . . . 6 |- ( n = d -> ( A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) <-> A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) ) )
22	21	rexbidv 2478	. . . . 5 |- ( n = d -> ( E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) <-> E. e e. NN0 A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) ) )
23	22	cbvralv 2705	. . . 4 |- ( A. n e. NN0 E. e e. NN0 A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) <-> A. d e. NN0 E. e e. NN0 A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) )
24	19, 23	bitri 184	. . 3 |- ( A. n e. NN0 E. k e. NN0 A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> A. d e. NN0 E. e e. NN0 A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) )
25	9, 24	sylib 122	. 2 |- ( ph -> A. d e. NN0 E. e e. NN0 A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) )
26		oveq1 5885	. . . 4 |- ( c = a -> ( c + 1 ) = ( a + 1 ) )
27	26	cbvmptv 4101	. . 3 |- ( c e. ZZ |-> ( c + 1 ) ) = ( a e. ZZ |-> ( a + 1 ) )
28		freceq1 6396	. . 3 |- ( ( c e. ZZ |-> ( c + 1 ) ) = ( a e. ZZ |-> ( a + 1 ) ) -> frec ( ( c e. ZZ |-> ( c + 1 ) ) , 0 ) = frec ( ( a e. ZZ |-> ( a + 1 ) ) , 0 ) )
29	27, 28	ax-mp 5	. 2 |- frec ( ( c e. ZZ |-> ( c + 1 ) ) , 0 ) = frec ( ( a e. ZZ |-> ( a + 1 ) ) , 0 )
30	7, 8, 25, 29	ennnfonelemnn0 12426	1 |- ( ph -> A ~~ NN )

Syntax hints:   -> wi 4  DECID wdc 834   = wceq 1353   =/= wne 2347   A.wral 2455   E.wrex 2456   class class class wbr 4005   |-> cmpt 4066   -onto->wfo 5216   ` cfv 5218  (class class class)co 5878  freccfrec 6394   ~~ cen 6741   0cc0 7814   1c1 7815   + caddc 7817   NNcn 8922   NN0cn0 9179   ZZcz 9256   ...cfz 10011

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-addcom 7914  ax-addass 7916  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-0id 7922  ax-rnegex 7923  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-ltadd 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-er 6538  df-pm 6654  df-en 6744  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-inn 8923  df-n0 9180  df-z 9257  df-uz 9532  df-fz 10012  df-seqfrec 10449

This theorem is referenced by:  ennnfone  12429