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Theorem reff 30237
Description: For any cover refinement, there exists a function associating with each set in the refinement a set in the original cover containing it. This is sometimes used as a defintion of refinement. Note that this definition uses the axiom of choice through ac6sg 9523. (Contributed by Thierry Arnoux, 12-Jan-2020.)
Assertion
Ref Expression
reff (𝐴𝑉 → (𝐴Ref𝐵 ↔ ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))))
Distinct variable groups:   𝐴,𝑓,𝑣   𝐵,𝑓,𝑣   𝑓,𝑉,𝑣

Proof of Theorem reff
Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3766 . . . 4 𝐵 𝐵
2 eqid 2761 . . . . . 6 𝐴 = 𝐴
3 eqid 2761 . . . . . 6 𝐵 = 𝐵
42, 3isref 21535 . . . . 5 (𝐴𝑉 → (𝐴Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑣𝐴𝑢𝐵 𝑣𝑢)))
54simprbda 654 . . . 4 ((𝐴𝑉𝐴Ref𝐵) → 𝐵 = 𝐴)
61, 5syl5sseq 3795 . . 3 ((𝐴𝑉𝐴Ref𝐵) → 𝐵 𝐴)
74simplbda 655 . . . 4 ((𝐴𝑉𝐴Ref𝐵) → ∀𝑣𝐴𝑢𝐵 𝑣𝑢)
8 sseq2 3769 . . . . . 6 (𝑢 = (𝑓𝑣) → (𝑣𝑢𝑣 ⊆ (𝑓𝑣)))
98ac6sg 9523 . . . . 5 (𝐴𝑉 → (∀𝑣𝐴𝑢𝐵 𝑣𝑢 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))))
109adantr 472 . . . 4 ((𝐴𝑉𝐴Ref𝐵) → (∀𝑣𝐴𝑢𝐵 𝑣𝑢 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))))
117, 10mpd 15 . . 3 ((𝐴𝑉𝐴Ref𝐵) → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))
126, 11jca 555 . 2 ((𝐴𝑉𝐴Ref𝐵) → ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))))
13 simplr 809 . . . . . . 7 (((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) → 𝐵 𝐴)
14 nfv 1993 . . . . . . . . . . . . 13 𝑣(𝐴𝑉 𝐵 𝐴)
15 nfv 1993 . . . . . . . . . . . . . 14 𝑣 𝑓:𝐴𝐵
16 nfra1 3080 . . . . . . . . . . . . . 14 𝑣𝑣𝐴 𝑣 ⊆ (𝑓𝑣)
1715, 16nfan 1978 . . . . . . . . . . . . 13 𝑣(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))
1814, 17nfan 1978 . . . . . . . . . . . 12 𝑣((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))
19 nfv 1993 . . . . . . . . . . . 12 𝑣 𝑥 𝐴
2018, 19nfan 1978 . . . . . . . . . . 11 𝑣(((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴)
21 simplrl 819 . . . . . . . . . . . . . . 15 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑣𝐴) → 𝑓:𝐴𝐵)
22 simpr 479 . . . . . . . . . . . . . . 15 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑣𝐴) → 𝑣𝐴)
2321, 22ffvelrnd 6525 . . . . . . . . . . . . . 14 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑣𝐴) → (𝑓𝑣) ∈ 𝐵)
2423adantlr 753 . . . . . . . . . . . . 13 (((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) ∧ 𝑣𝐴) → (𝑓𝑣) ∈ 𝐵)
2524adantr 472 . . . . . . . . . . . 12 ((((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) ∧ 𝑣𝐴) ∧ 𝑥𝑣) → (𝑓𝑣) ∈ 𝐵)
26 simplrr 820 . . . . . . . . . . . . . . 15 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑣𝐴) → ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))
2726adantlr 753 . . . . . . . . . . . . . 14 (((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) ∧ 𝑣𝐴) → ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))
28 simpr 479 . . . . . . . . . . . . . 14 (((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) ∧ 𝑣𝐴) → 𝑣𝐴)
29 rspa 3069 . . . . . . . . . . . . . 14 ((∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣) ∧ 𝑣𝐴) → 𝑣 ⊆ (𝑓𝑣))
3027, 28, 29syl2anc 696 . . . . . . . . . . . . 13 (((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) ∧ 𝑣𝐴) → 𝑣 ⊆ (𝑓𝑣))
3130sselda 3745 . . . . . . . . . . . 12 ((((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) ∧ 𝑣𝐴) ∧ 𝑥𝑣) → 𝑥 ∈ (𝑓𝑣))
32 eleq2 2829 . . . . . . . . . . . . 13 (𝑢 = (𝑓𝑣) → (𝑥𝑢𝑥 ∈ (𝑓𝑣)))
3332rspcev 3450 . . . . . . . . . . . 12 (((𝑓𝑣) ∈ 𝐵𝑥 ∈ (𝑓𝑣)) → ∃𝑢𝐵 𝑥𝑢)
3425, 31, 33syl2anc 696 . . . . . . . . . . 11 ((((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) ∧ 𝑣𝐴) ∧ 𝑥𝑣) → ∃𝑢𝐵 𝑥𝑢)
35 simpr 479 . . . . . . . . . . . 12 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) → 𝑥 𝐴)
36 eluni2 4593 . . . . . . . . . . . 12 (𝑥 𝐴 ↔ ∃𝑣𝐴 𝑥𝑣)
3735, 36sylib 208 . . . . . . . . . . 11 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) → ∃𝑣𝐴 𝑥𝑣)
3820, 34, 37r19.29af 3215 . . . . . . . . . 10 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) → ∃𝑢𝐵 𝑥𝑢)
39 eluni2 4593 . . . . . . . . . 10 (𝑥 𝐵 ↔ ∃𝑢𝐵 𝑥𝑢)
4038, 39sylibr 224 . . . . . . . . 9 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑥 𝐴) → 𝑥 𝐵)
4140ex 449 . . . . . . . 8 (((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) → (𝑥 𝐴𝑥 𝐵))
4241ssrdv 3751 . . . . . . 7 (((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) → 𝐴 𝐵)
4313, 42eqssd 3762 . . . . . 6 (((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) → 𝐵 = 𝐴)
4426, 22, 29syl2anc 696 . . . . . . . . 9 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑣𝐴) → 𝑣 ⊆ (𝑓𝑣))
458rspcev 3450 . . . . . . . . 9 (((𝑓𝑣) ∈ 𝐵𝑣 ⊆ (𝑓𝑣)) → ∃𝑢𝐵 𝑣𝑢)
4623, 44, 45syl2anc 696 . . . . . . . 8 ((((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) ∧ 𝑣𝐴) → ∃𝑢𝐵 𝑣𝑢)
4746ex 449 . . . . . . 7 (((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) → (𝑣𝐴 → ∃𝑢𝐵 𝑣𝑢))
4818, 47ralrimi 3096 . . . . . 6 (((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) → ∀𝑣𝐴𝑢𝐵 𝑣𝑢)
494ad2antrr 764 . . . . . 6 (((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) → (𝐴Ref𝐵 ↔ ( 𝐵 = 𝐴 ∧ ∀𝑣𝐴𝑢𝐵 𝑣𝑢)))
5043, 48, 49mpbir2and 995 . . . . 5 (((𝐴𝑉 𝐵 𝐴) ∧ (𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣))) → 𝐴Ref𝐵)
5150ex 449 . . . 4 ((𝐴𝑉 𝐵 𝐴) → ((𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)) → 𝐴Ref𝐵))
5251exlimdv 2011 . . 3 ((𝐴𝑉 𝐵 𝐴) → (∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)) → 𝐴Ref𝐵))
5352impr 650 . 2 ((𝐴𝑉 ∧ ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))) → 𝐴Ref𝐵)
5412, 53impbida 913 1 (𝐴𝑉 → (𝐴Ref𝐵 ↔ ( 𝐵 𝐴 ∧ ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑣𝐴 𝑣 ⊆ (𝑓𝑣)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2140  wral 3051  wrex 3052  wss 3716   cuni 4589   class class class wbr 4805  wf 6046  cfv 6050  Refcref 21528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-reg 8665  ax-inf2 8714  ax-ac2 9498
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-se 5227  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-isom 6059  df-riota 6776  df-om 7233  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-en 8125  df-r1 8803  df-rank 8804  df-card 8976  df-ac 9150  df-ref 21531
This theorem is referenced by:  locfinreflem  30238
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