Theorem ennnfonelemr 12437

Description: Lemma for ennnfone 12439. 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 1722	. . . . 5 |- ( x = a -> ( x = y <-> a = y ) )
3	2	dcbid 839	. . . 4 |- ( x = a -> (DECID x = y <-> DECID a = y ) )
4		equequ2 1723	. . . . 5 |- ( y = b -> ( a = y <-> a = b ) )
5	4	dcbid 839	. . . 4 |- ( y = b -> (DECID a = y <-> DECID a = b ) )
6	3, 5	cbvral2v 2728	. . 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 5527	. . . . . . . . 9 |- ( j = f -> ( F ` j ) = ( F ` f ) )
11	10	neeq2d 2376	. . . . . . . 8 |- ( j = f -> ( ( F ` k ) =/= ( F ` j ) <-> ( F ` k ) =/= ( F ` f ) ) )
12	11	cbvralv 2715	. . . . . . 7 |- ( A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) )
13	12	rexbii 2494	. . . . . 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 5527	. . . . . . . . 9 |- ( k = e -> ( F ` k ) = ( F ` e ) )
15	14	neeq1d 2375	. . . . . . . 8 |- ( k = e -> ( ( F ` k ) =/= ( F ` f ) <-> ( F ` e ) =/= ( F ` f ) ) )
16	15	ralbidv 2487	. . . . . . 7 |- ( k = e -> ( A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) <-> A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) ) )
17	16	cbvrexv 2716	. . . . . 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 2493	. . . 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 5896	. . . . . . 7 |- ( n = d -> ( 0 ... n ) = ( 0 ... d ) )
21	20	raleqdv 2689	. . . . . 6 |- ( n = d -> ( A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) <-> A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) ) )
22	21	rexbidv 2488	. . . . 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 2715	. . . 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 5895	. . . 4 |- ( c = a -> ( c + 1 ) = ( a + 1 ) )
27	26	cbvmptv 4111	. . 3 |- ( c e. ZZ |-> ( c + 1 ) ) = ( a e. ZZ |-> ( a + 1 ) )
28		freceq1 6406	. . 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 12436	1 |- ( ph -> A ~~ NN )

Syntax hints:   -> wi 4  DECID wdc 835   = wceq 1363   =/= wne 2357   A.wral 2465   E.wrex 2466   class class class wbr 4015   |-> cmpt 4076   -onto->wfo 5226   ` cfv 5228  (class class class)co 5888  freccfrec 6404   ~~ cen 6751   0cc0 7824   1c1 7825   + caddc 7827   NNcn 8932   NN0cn0 9189   ZZcz 9266   ...cfz 10021

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-er 6548  df-pm 6664  df-en 6754  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542  df-fz 10022  df-seqfrec 10459

This theorem is referenced by:  ennnfone  12439