Theorem birthdaylem1g 16070

Description: Lemma for birthdaylog2 16073. (Contributed by Mario Carneiro, 17-Apr-2015.)

Hypotheses

Ref	Expression
birthday.s	|- S = { f | f : ( 1 ... K ) --> ( 1 ... N ) }
birthday.t	|- T = { f | f : ( 1 ... K ) -1-1-> ( 1 ... N ) }

Assertion

Ref	Expression
birthdaylem1g	|- ( ( K e. NN0 /\ N e. NN ) -> ( T C_ S /\ S e. Fin /\ S =/= (/) ) )

Distinct variable groups:   f, K   f, N

Allowed substitution hints:   S( f)   T( f)

Proof of Theorem birthdaylem1g

Step	Hyp	Ref	Expression
1		f1f 5596	. . . . 5 |- ( f : ( 1 ... K ) -1-1-> ( 1 ... N ) -> f : ( 1 ... K ) --> ( 1 ... N ) )
2	1	ss2abi 3320	. . . 4 |- { f | f : ( 1 ... K ) -1-1-> ( 1 ... N ) } C_ { f | f : ( 1 ... K ) --> ( 1 ... N ) }
3		birthday.t	. . . 4 |- T = { f | f : ( 1 ... K ) -1-1-> ( 1 ... N ) }
4		birthday.s	. . . 4 |- S = { f | f : ( 1 ... K ) --> ( 1 ... N ) }
5	2, 3, 4	3sstr4i 3289	. . 3 |- T C_ S
6	5	a1i 9	. 2 |- ( ( K e. NN0 /\ N e. NN ) -> T C_ S )
7		1zzd 9654	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> 1 e. ZZ )
8		nnz 9646	. . . . . . 7 |- ( N e. NN -> N e. ZZ )
9	8	adantl 277	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> N e. ZZ )
10	7, 9	fzfigd 10851	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( 1 ... N ) e. Fin )
11		nn0z 9647	. . . . . . 7 |- ( K e. NN0 -> K e. ZZ )
12	11	adantr 276	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> K e. ZZ )
13	7, 12	fzfigd 10851	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( 1 ... K ) e. Fin )
14		mapvalg 6926	. . . . 5 |- ( ( ( 1 ... N ) e. Fin /\ ( 1 ... K ) e. Fin ) -> ( ( 1 ... N ) ^m ( 1 ... K ) ) = { f | f : ( 1 ... K ) --> ( 1 ... N ) } )
15	10, 13, 14	syl2anc 415	. . . 4 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( 1 ... N ) ^m ( 1 ... K ) ) = { f | f : ( 1 ... K ) --> ( 1 ... N ) } )
16	4, 15	eqtr4id 2290	. . 3 |- ( ( K e. NN0 /\ N e. NN ) -> S = ( ( 1 ... N ) ^m ( 1 ... K ) ) )
17		mapfi 7255	. . . 4 |- ( ( ( 1 ... N ) e. Fin /\ ( 1 ... K ) e. Fin ) -> ( ( 1 ... N ) ^m ( 1 ... K ) ) e. Fin )
18	10, 13, 17	syl2anc 415	. . 3 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( 1 ... N ) ^m ( 1 ... K ) ) e. Fin )
19	16, 18	eqeltrd 2315	. 2 |- ( ( K e. NN0 /\ N e. NN ) -> S e. Fin )
20		elfz1end 10444	. . . . 5 |- ( N e. NN <-> N e. ( 1 ... N ) )
21		ne0i 3528	. . . . 5 |- ( N e. ( 1 ... N ) -> ( 1 ... N ) =/= (/) )
22	20, 21	sylbi 121	. . . 4 |- ( N e. NN -> ( 1 ... N ) =/= (/) )
23	22	adantl 277	. . 3 |- ( ( K e. NN0 /\ N e. NN ) -> ( 1 ... N ) =/= (/) )
24	16	eqeq1d 2247	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( S = (/) <-> ( ( 1 ... N ) ^m ( 1 ... K ) ) = (/) ) )
25		map0g 6963	. . . . . . 7 |- ( ( ( 1 ... N ) e. Fin /\ ( 1 ... K ) e. Fin ) -> ( ( ( 1 ... N ) ^m ( 1 ... K ) ) = (/) <-> ( ( 1 ... N ) = (/) /\ ( 1 ... K ) =/= (/) ) ) )
26	10, 13, 25	syl2anc 415	. . . . . 6 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( ( 1 ... N ) ^m ( 1 ... K ) ) = (/) <-> ( ( 1 ... N ) = (/) /\ ( 1 ... K ) =/= (/) ) ) )
27		simpl 109	. . . . . 6 |- ( ( ( 1 ... N ) = (/) /\ ( 1 ... K ) =/= (/) ) -> ( 1 ... N ) = (/) )
28	26, 27	biimtrdi 163	. . . . 5 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( ( 1 ... N ) ^m ( 1 ... K ) ) = (/) -> ( 1 ... N ) = (/) ) )
29	24, 28	sylbid 150	. . . 4 |- ( ( K e. NN0 /\ N e. NN ) -> ( S = (/) -> ( 1 ... N ) = (/) ) )
30	29	necon3d 2464	. . 3 |- ( ( K e. NN0 /\ N e. NN ) -> ( ( 1 ... N ) =/= (/) -> S =/= (/) ) )
31	23, 30	mpd 13	. 2 |- ( ( K e. NN0 /\ N e. NN ) -> S =/= (/) )
32	6, 19, 31	3jca 1208	1 |- ( ( K e. NN0 /\ N e. NN ) -> ( T C_ S /\ S e. Fin /\ S =/= (/) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   =/= wne 2420   C_ wss 3220   (/)c0 3520   -->wf 5371   -1-1->wf1 5372  (class class class)co 6079   ^m cmap 6916   Fincfn 7016   1c1 8174   NNcn 9287   NN0cn0 9546   ZZcz 9627   ...cfz 10394

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-1o 6681  df-er 6801  df-map 6918  df-en 7017  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-fz 10395

This theorem is referenced by:  birthdaylem3  16072  birthdaylog2  16073