Theorem acsmap2d 17179

Description: In an algebraic closure system, if S and T have the same closure and S is independent, then there is a map f from T into the set of finite subsets of S such that S equals the union of ran f. This is proven by taking the map f from acsmapd 17178 and observing that, since S and T have the same closure, the closure of U. ran f must contain S. Since S is independent, by mrissmrcd 16300, U. ran f must equal S. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
acsmap2d.1	|- ( ph -> A e. (ACS ` X ) )
acsmap2d.2	|- N = (mrCls ` A )
acsmap2d.3	|- I = (mrInd ` A )
acsmap2d.4	|- ( ph -> S e. I )
acsmap2d.5	|- ( ph -> T C_ X )
acsmap2d.6	|- ( ph -> ( N ` S ) = ( N ` T ) )

Assertion

Ref	Expression
acsmap2d	|- ( ph -> E. f ( f : T --> ( ~P S i^i Fin ) /\ S = U. ran f ) )

Distinct variable groups:   S, f   T, f   ph, f   f, N

Allowed substitution hints:   A( f)   I( f)   X( f)

Proof of Theorem acsmap2d

Step	Hyp	Ref	Expression
1		acsmap2d.1	. . 3 |- ( ph -> A e. (ACS ` X ) )
2		acsmap2d.2	. . 3 |- N = (mrCls ` A )
3		acsmap2d.3	. . . 4 |- I = (mrInd ` A )
4	1	acsmred 16317	. . . 4 |- ( ph -> A e. (Moore ` X ) )
5		acsmap2d.4	. . . 4 |- ( ph -> S e. I )
6	3, 4, 5	mrissd 16296	. . 3 |- ( ph -> S C_ X )
7		acsmap2d.5	. . . . 5 |- ( ph -> T C_ X )
8	4, 2, 7	mrcssidd 16285	. . . 4 |- ( ph -> T C_ ( N ` T ) )
9		acsmap2d.6	. . . 4 |- ( ph -> ( N ` S ) = ( N ` T ) )
10	8, 9	sseqtr4d 3642	. . 3 |- ( ph -> T C_ ( N ` S ) )
11	1, 2, 6, 10	acsmapd 17178	. 2 |- ( ph -> E. f ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) )
12		simprl 794	. . . . 5 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> f : T --> ( ~P S i^i Fin ) )
13	4	adantr 481	. . . . . 6 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> A e. (Moore ` X ) )
14	5	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> S e. I )
15	3, 13, 14	mrissd 16296	. . . . . . . 8 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> S C_ X )
16	13, 2, 15	mrcssidd 16285	. . . . . . 7 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> S C_ ( N ` S ) )
17	9	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> ( N ` S ) = ( N ` T ) )
18		simprr 796	. . . . . . . . . 10 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> T C_ ( N ` U. ran f ) )
19	13, 2	mrcssvd 16283	. . . . . . . . . 10 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> ( N ` U. ran f ) C_ X )
20	13, 2, 18, 19	mrcssd 16284	. . . . . . . . 9 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> ( N ` T ) C_ ( N ` ( N ` U. ran f ) ) )
21		frn 6053	. . . . . . . . . . . . . 14 |- ( f : T --> ( ~P S i^i Fin ) -> ran f C_ ( ~P S i^i Fin ) )
22	21	unissd 4462	. . . . . . . . . . . . 13 |- ( f : T --> ( ~P S i^i Fin ) -> U. ran f C_ U. ( ~P S i^i Fin ) )
23		unifpw 8269	. . . . . . . . . . . . 13 |- U. ( ~P S i^i Fin ) = S
24	22, 23	syl6sseq 3651	. . . . . . . . . . . 12 |- ( f : T --> ( ~P S i^i Fin ) -> U. ran f C_ S )
25	24	ad2antrl 764	. . . . . . . . . . 11 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> U. ran f C_ S )
26	25, 15	sstrd 3613	. . . . . . . . . 10 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> U. ran f C_ X )
27	13, 2, 26	mrcidmd 16286	. . . . . . . . 9 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> ( N ` ( N ` U. ran f ) ) = ( N ` U. ran f ) )
28	20, 27	sseqtrd 3641	. . . . . . . 8 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> ( N ` T ) C_ ( N ` U. ran f ) )
29	17, 28	eqsstrd 3639	. . . . . . 7 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> ( N ` S ) C_ ( N ` U. ran f ) )
30	16, 29	sstrd 3613	. . . . . 6 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> S C_ ( N ` U. ran f ) )
31	13, 2, 3, 30, 25, 14	mrissmrcd 16300	. . . . 5 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> S = U. ran f )
32	12, 31	jca 554	. . . 4 |- ( ( ph /\ ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) ) -> ( f : T --> ( ~P S i^i Fin ) /\ S = U. ran f ) )
33	32	ex 450	. . 3 |- ( ph -> ( ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) -> ( f : T --> ( ~P S i^i Fin ) /\ S = U. ran f ) ) )
34	33	eximdv 1846	. 2 |- ( ph -> ( E. f ( f : T --> ( ~P S i^i Fin ) /\ T C_ ( N ` U. ran f ) ) -> E. f ( f : T --> ( ~P S i^i Fin ) /\ S = U. ran f ) ) )
35	11, 34	mpd 15	1 |- ( ph -> E. f ( f : T --> ( ~P S i^i Fin ) /\ S = U. ran f ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   ran crn 5115   -->wf 5884   ` cfv 5888   Fincfn 7955  Moorecmre 16242  mrClscmrc 16243  mrIndcmri 16244  ACScacs 16245

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538  ax-ac2 9285  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-int 4476  df-iun 4522  df-iin 4523  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-se 5074  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-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-r1 8627  df-rank 8628  df-card 8765  df-ac 8939  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-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-tset 15960  df-ple 15961  df-ocomp 15963  df-mre 16246  df-mrc 16247  df-mri 16248  df-acs 16249  df-preset 16928  df-drs 16929  df-poset 16946  df-ipo 17152

This theorem is referenced by:  acsinfd  17180  acsdomd  17181