Theorem ennnfonelemr 12640

Description: Lemma for ennnfone 12642. 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 1726	. . . . 5 |- ( x = a -> ( x = y <-> a = y ) )
3	2	dcbid 839	. . . 4 |- ( x = a -> (DECID x = y <-> DECID a = y ) )
4		equequ2 1727	. . . . 5 |- ( y = b -> ( a = y <-> a = b ) )
5	4	dcbid 839	. . . 4 |- ( y = b -> (DECID a = y <-> DECID a = b ) )
6	3, 5	cbvral2v 2742	. . 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 5558	. . . . . . . . 9 |- ( j = f -> ( F ` j ) = ( F ` f ) )
11	10	neeq2d 2386	. . . . . . . 8 |- ( j = f -> ( ( F ` k ) =/= ( F ` j ) <-> ( F ` k ) =/= ( F ` f ) ) )
12	11	cbvralv 2729	. . . . . . 7 |- ( A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) )
13	12	rexbii 2504	. . . . . 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 5558	. . . . . . . . 9 |- ( k = e -> ( F ` k ) = ( F ` e ) )
15	14	neeq1d 2385	. . . . . . . 8 |- ( k = e -> ( ( F ` k ) =/= ( F ` f ) <-> ( F ` e ) =/= ( F ` f ) ) )
16	15	ralbidv 2497	. . . . . . 7 |- ( k = e -> ( A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) <-> A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) ) )
17	16	cbvrexv 2730	. . . . . 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 2503	. . . 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 5930	. . . . . . 7 |- ( n = d -> ( 0 ... n ) = ( 0 ... d ) )
21	20	raleqdv 2699	. . . . . 6 |- ( n = d -> ( A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) <-> A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) ) )
22	21	rexbidv 2498	. . . . 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 2729	. . . 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 5929	. . . 4 |- ( c = a -> ( c + 1 ) = ( a + 1 ) )
27	26	cbvmptv 4129	. . 3 |- ( c e. ZZ |-> ( c + 1 ) ) = ( a e. ZZ |-> ( a + 1 ) )
28		freceq1 6450	. . 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 12639	1 |- ( ph -> A ~~ NN )

Syntax hints:   -> wi 4  DECID wdc 835   = wceq 1364   =/= wne 2367   A.wral 2475   E.wrex 2476   class class class wbr 4033   |-> cmpt 4094   -onto->wfo 5256   ` cfv 5258  (class class class)co 5922  freccfrec 6448   ~~ cen 6797   0cc0 7879   1c1 7880   + caddc 7882   NNcn 8990   NN0cn0 9249   ZZcz 9326   ...cfz 10083

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-er 6592  df-pm 6710  df-en 6800  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-seqfrec 10540

This theorem is referenced by:  ennnfone  12642