Theorem ennnfonelemr 12827

Description: Lemma for ennnfone 12829. 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 1735	. . . . 5 |- ( x = a -> ( x = y <-> a = y ) )
3	2	dcbid 840	. . . 4 |- ( x = a -> (DECID x = y <-> DECID a = y ) )
4		equequ2 1736	. . . . 5 |- ( y = b -> ( a = y <-> a = b ) )
5	4	dcbid 840	. . . 4 |- ( y = b -> (DECID a = y <-> DECID a = b ) )
6	3, 5	cbvral2v 2751	. . 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 5578	. . . . . . . . 9 |- ( j = f -> ( F ` j ) = ( F ` f ) )
11	10	neeq2d 2395	. . . . . . . 8 |- ( j = f -> ( ( F ` k ) =/= ( F ` j ) <-> ( F ` k ) =/= ( F ` f ) ) )
12	11	cbvralv 2738	. . . . . . 7 |- ( A. j e. ( 0 ... n ) ( F ` k ) =/= ( F ` j ) <-> A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) )
13	12	rexbii 2513	. . . . . 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 5578	. . . . . . . . 9 |- ( k = e -> ( F ` k ) = ( F ` e ) )
15	14	neeq1d 2394	. . . . . . . 8 |- ( k = e -> ( ( F ` k ) =/= ( F ` f ) <-> ( F ` e ) =/= ( F ` f ) ) )
16	15	ralbidv 2506	. . . . . . 7 |- ( k = e -> ( A. f e. ( 0 ... n ) ( F ` k ) =/= ( F ` f ) <-> A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) ) )
17	16	cbvrexv 2739	. . . . . 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 2512	. . . 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 5954	. . . . . . 7 |- ( n = d -> ( 0 ... n ) = ( 0 ... d ) )
21	20	raleqdv 2708	. . . . . 6 |- ( n = d -> ( A. f e. ( 0 ... n ) ( F ` e ) =/= ( F ` f ) <-> A. f e. ( 0 ... d ) ( F ` e ) =/= ( F ` f ) ) )
22	21	rexbidv 2507	. . . . 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 2738	. . . 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 5953	. . . 4 |- ( c = a -> ( c + 1 ) = ( a + 1 ) )
27	26	cbvmptv 4141	. . 3 |- ( c e. ZZ |-> ( c + 1 ) ) = ( a e. ZZ |-> ( a + 1 ) )
28		freceq1 6480	. . 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 12826	1 |- ( ph -> A ~~ NN )

Syntax hints:   -> wi 4  DECID wdc 836   = wceq 1373   =/= wne 2376   A.wral 2484   E.wrex 2485   class class class wbr 4045   |-> cmpt 4106   -onto->wfo 5270   ` cfv 5272  (class class class)co 5946  freccfrec 6478   ~~ cen 6827   0cc0 7927   1c1 7928   + caddc 7930   NNcn 9038   NN0cn0 9297   ZZcz 9374   ...cfz 10132

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-ltadd 8043

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-recs 6393  df-frec 6479  df-er 6622  df-pm 6740  df-en 6830  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-inn 9039  df-n0 9298  df-z 9375  df-uz 9651  df-fz 10133  df-seqfrec 10595

This theorem is referenced by:  ennnfone  12829