Theorem acsmap2d 17606

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 17605 and observing that, since 𝑆 and 𝑇 have the same closure, the closure of ∪ ran 𝑓 must contain 𝑆. Since 𝑆 is independent, by mrissmrcd 16728, ∪ 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	⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))
acsmap2d.2	⊢ 𝑁 = (mrCls‘𝐴)
acsmap2d.3	⊢ 𝐼 = (mrInd‘𝐴)
acsmap2d.4	⊢ (𝜑 → 𝑆 ∈ 𝐼)
acsmap2d.5	⊢ (𝜑 → 𝑇 ⊆ 𝑋)
acsmap2d.6	⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))

Assertion

Ref	Expression
acsmap2d	⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))

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

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

Proof of Theorem acsmap2d

Step	Hyp	Ref	Expression
1		acsmap2d.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))
2		acsmap2d.2	. . 3 ⊢ 𝑁 = (mrCls‘𝐴)
3		acsmap2d.3	. . . 4 ⊢ 𝐼 = (mrInd‘𝐴)
4	1	acsmred 16744	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋))
5		acsmap2d.4	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐼)
6	3, 4, 5	mrissd 16724	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑋)
7		acsmap2d.5	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ 𝑋)
8	4, 2, 7	mrcssidd 16713	. . . 4 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑇))
9		acsmap2d.6	. . . 4 ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))
10	8, 9	sseqtr4d 3924	. . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆))
11	1, 2, 6, 10	acsmapd 17605	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)))
12		simprl 767	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin))
13	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝐴 ∈ (Moore‘𝑋))
14	5	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ∈ 𝐼)
15	3, 13, 14	mrissd 16724	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ 𝑋)
16	13, 2, 15	mrcssidd 16713	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ (𝑁‘𝑆))
17	9	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑆) = (𝑁‘𝑇))
18		simprr 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑇 ⊆ (𝑁‘∪ ran 𝑓))
19	13, 2	mrcssvd 16711	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘∪ ran 𝑓) ⊆ 𝑋)
20	13, 2, 18, 19	mrcssd 16712	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑇) ⊆ (𝑁‘(𝑁‘∪ ran 𝑓)))
21		frn 6380	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ran 𝑓 ⊆ (𝒫 𝑆 ∩ Fin))
22	21	unissd 4763	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ∪ ran 𝑓 ⊆ ∪ (𝒫 𝑆 ∩ Fin))
23		unifpw 8663	. . . . . . . . . . . . 13 ⊢ ∪ (𝒫 𝑆 ∩ Fin) = 𝑆
24	22, 23	syl6sseq 3933	. . . . . . . . . . . 12 ⊢ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) → ∪ ran 𝑓 ⊆ 𝑆)
25	24	ad2antrl 724	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → ∪ ran 𝑓 ⊆ 𝑆)
26	25, 15	sstrd 3894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → ∪ ran 𝑓 ⊆ 𝑋)
27	13, 2, 26	mrcidmd 16714	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘(𝑁‘∪ ran 𝑓)) = (𝑁‘∪ ran 𝑓))
28	20, 27	sseqtrd 3923	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑇) ⊆ (𝑁‘∪ ran 𝑓))
29	17, 28	eqsstrd 3921	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑁‘𝑆) ⊆ (𝑁‘∪ ran 𝑓))
30	16, 29	sstrd 3894	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 ⊆ (𝑁‘∪ ran 𝑓))
31	13, 2, 3, 30, 25, 14	mrissmrcd 16728	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → 𝑆 = ∪ ran 𝑓)
32	12, 31	jca 512	. . . 4 ⊢ ((𝜑 ∧ (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓))) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))
33	32	ex 413	. . 3 ⊢ (𝜑 → ((𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)) → (𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)))
34	33	eximdv 1893	. 2 ⊢ (𝜑 → (∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)) → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓)))
35	11, 34	mpd 15	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1520  ∃wex 1759   ∈ wcel 2079   ∩ cin 3853   ⊆ wss 3854  𝒫 cpw 4447  ∪ cuni 4739  ran crn 5436  ⟶wf 6213  ‘cfv 6217  Fincfn 8347  Moorecmre 16670  mrClscmrc 16671  mrIndcmri 16672  ACScacs 16673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-reg 8892  ax-inf2 8939  ax-ac2 9720  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-iin 4822  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-se 5395  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-isom 6226  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-oadd 7948  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-r1 9028  df-rank 9029  df-card 9203  df-ac 9377  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-5 11540  df-6 11541  df-7 11542  df-8 11543  df-9 11544  df-n0 11735  df-z 11819  df-dec 11937  df-uz 12083  df-fz 12732  df-struct 16302  df-ndx 16303  df-slot 16304  df-base 16306  df-tset 16401  df-ple 16402  df-ocomp 16403  df-mre 16674  df-mrc 16675  df-mri 16676  df-acs 16677  df-proset 17355  df-drs 17356  df-poset 17373  df-ipo 17579

This theorem is referenced by:  acsinfd  17607  acsdomd  17608