Theorem ssnnf1octb 39382

Description: There exists a bijection between a subset of NN and a given nonempty countable set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Assertion

Ref	Expression
ssnnf1octb	|- ( ( A ~<_ om /\ A =/= (/) ) -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) )

Distinct variable group:   A, f

Proof of Theorem ssnnf1octb

Dummy variables g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnfoctb 39213	. 2 |- ( ( A ~<_ om /\ A =/= (/) ) -> E. g g : NN -onto-> A )
2		fofn 6117	. . . . . 6 |- ( g : NN -onto-> A -> g Fn NN )
3		nnex 11026	. . . . . . 7 |- NN e. _V
4	3	a1i 11	. . . . . 6 |- ( g : NN -onto-> A -> NN e. _V )
5		ltwenn 12761	. . . . . . 7 |- < We NN
6	5	a1i 11	. . . . . 6 |- ( g : NN -onto-> A -> < We NN )
7	2, 4, 6	wessf1orn 39372	. . . . 5 |- ( g : NN -onto-> A -> E. x e. ~P NN ( g |` x ) : x -1-1-onto-> ran g )
8		f1odm 6141	. . . . . . . . . . 11 |- ( ( g |` x ) : x -1-1-onto-> ran g -> dom ( g |` x ) = x )
9	8	adantl 482	. . . . . . . . . 10 |- ( ( x e. ~P NN /\ ( g |` x ) : x -1-1-onto-> ran g ) -> dom ( g |` x ) = x )
10		elpwi 4168	. . . . . . . . . . 11 |- ( x e. ~P NN -> x C_ NN )
11	10	adantr 481	. . . . . . . . . 10 |- ( ( x e. ~P NN /\ ( g |` x ) : x -1-1-onto-> ran g ) -> x C_ NN )
12	9, 11	eqsstrd 3639	. . . . . . . . 9 |- ( ( x e. ~P NN /\ ( g |` x ) : x -1-1-onto-> ran g ) -> dom ( g |` x ) C_ NN )
13	12	3adant1 1079	. . . . . . . 8 |- ( ( g : NN -onto-> A /\ x e. ~P NN /\ ( g |` x ) : x -1-1-onto-> ran g ) -> dom ( g |` x ) C_ NN )
14		simpr 477	. . . . . . . . . 10 |- ( ( g : NN -onto-> A /\ ( g |` x ) : x -1-1-onto-> ran g ) -> ( g |` x ) : x -1-1-onto-> ran g )
15		eqidd 2623	. . . . . . . . . . 11 |- ( ( g : NN -onto-> A /\ ( g |` x ) : x -1-1-onto-> ran g ) -> ( g |` x ) = ( g |` x ) )
16	8	eqcomd 2628	. . . . . . . . . . . 12 |- ( ( g |` x ) : x -1-1-onto-> ran g -> x = dom ( g |` x ) )
17	16	adantl 482	. . . . . . . . . . 11 |- ( ( g : NN -onto-> A /\ ( g |` x ) : x -1-1-onto-> ran g ) -> x = dom ( g |` x ) )
18		forn 6118	. . . . . . . . . . . 12 |- ( g : NN -onto-> A -> ran g = A )
19	18	adantr 481	. . . . . . . . . . 11 |- ( ( g : NN -onto-> A /\ ( g |` x ) : x -1-1-onto-> ran g ) -> ran g = A )
20	15, 17, 19	f1oeq123d 6133	. . . . . . . . . 10 |- ( ( g : NN -onto-> A /\ ( g |` x ) : x -1-1-onto-> ran g ) -> ( ( g |` x ) : x -1-1-onto-> ran g <-> ( g |` x ) : dom ( g |` x ) -1-1-onto-> A ) )
21	14, 20	mpbid 222	. . . . . . . . 9 |- ( ( g : NN -onto-> A /\ ( g |` x ) : x -1-1-onto-> ran g ) -> ( g |` x ) : dom ( g |` x ) -1-1-onto-> A )
22	21	3adant2 1080	. . . . . . . 8 |- ( ( g : NN -onto-> A /\ x e. ~P NN /\ ( g |` x ) : x -1-1-onto-> ran g ) -> ( g |` x ) : dom ( g |` x ) -1-1-onto-> A )
23		vex 3203	. . . . . . . . . 10 |- g e. _V
24	23	resex 5443	. . . . . . . . 9 |- ( g |` x ) e. _V
25		dmeq 5324	. . . . . . . . . . 11 |- ( f = ( g |` x ) -> dom f = dom ( g |` x ) )
26	25	sseq1d 3632	. . . . . . . . . 10 |- ( f = ( g |` x ) -> ( dom f C_ NN <-> dom ( g |` x ) C_ NN ) )
27		id 22	. . . . . . . . . . 11 |- ( f = ( g |` x ) -> f = ( g |` x ) )
28		eqidd 2623	. . . . . . . . . . 11 |- ( f = ( g |` x ) -> A = A )
29	27, 25, 28	f1oeq123d 6133	. . . . . . . . . 10 |- ( f = ( g |` x ) -> ( f : dom f -1-1-onto-> A <-> ( g |` x ) : dom ( g |` x ) -1-1-onto-> A ) )
30	26, 29	anbi12d 747	. . . . . . . . 9 |- ( f = ( g |` x ) -> ( ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) <-> ( dom ( g |` x ) C_ NN /\ ( g |` x ) : dom ( g |` x ) -1-1-onto-> A ) ) )
31	24, 30	spcev 3300	. . . . . . . 8 |- ( ( dom ( g |` x ) C_ NN /\ ( g |` x ) : dom ( g |` x ) -1-1-onto-> A ) -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) )
32	13, 22, 31	syl2anc 693	. . . . . . 7 |- ( ( g : NN -onto-> A /\ x e. ~P NN /\ ( g |` x ) : x -1-1-onto-> ran g ) -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) )
33	32	3exp 1264	. . . . . 6 |- ( g : NN -onto-> A -> ( x e. ~P NN -> ( ( g |` x ) : x -1-1-onto-> ran g -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) ) ) )
34	33	rexlimdv 3030	. . . . 5 |- ( g : NN -onto-> A -> ( E. x e. ~P NN ( g |` x ) : x -1-1-onto-> ran g -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) ) )
35	7, 34	mpd 15	. . . 4 |- ( g : NN -onto-> A -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) )
36	35	a1i 11	. . 3 |- ( ( A ~<_ om /\ A =/= (/) ) -> ( g : NN -onto-> A -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) ) )
37	36	exlimdv 1861	. 2 |- ( ( A ~<_ om /\ A =/= (/) ) -> ( E. g g : NN -onto-> A -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) ) )
38	1, 37	mpd 15	1 |- ( ( A ~<_ om /\ A =/= (/) ) -> E. f ( dom f C_ NN /\ f : dom f -1-1-onto-> A ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   We wwe 5072   dom cdm 5114   ran crn 5115   |` cres 5116   -onto->wfo 5886   -1-1-onto->wf1o 5887   omcom 7065   ~<_ cdom 7953   < clt 10074   NNcn 11020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688

This theorem is referenced by:  isomennd  40745