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Theorem efgi2 18059
Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
efgval.r = ( ~FG𝐼)
efgval2.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
efgval2.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
Assertion
Ref Expression
efgi2 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → 𝐴 𝐵)
Distinct variable groups:   𝑦,𝑧   𝑣,𝑛,𝑤,𝑦,𝑧   𝑛,𝑀,𝑣,𝑤   𝑛,𝑊,𝑣,𝑤,𝑦,𝑧   𝑦, ,𝑧   𝑛,𝐼,𝑣,𝑤,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑧,𝑤,𝑣,𝑛)   𝐵(𝑦,𝑧,𝑤,𝑣,𝑛)   (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)

Proof of Theorem efgi2
Dummy variables 𝑎 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6148 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑇𝑎) = (𝑇𝐴))
21rneqd 5313 . . . . . . . . . 10 (𝑎 = 𝐴 → ran (𝑇𝑎) = ran (𝑇𝐴))
3 eceq1 7727 . . . . . . . . . 10 (𝑎 = 𝐴 → [𝑎]𝑟 = [𝐴]𝑟)
42, 3sseq12d 3613 . . . . . . . . 9 (𝑎 = 𝐴 → (ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇𝐴) ⊆ [𝐴]𝑟))
54rspcv 3291 . . . . . . . 8 (𝐴𝑊 → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 → ran (𝑇𝐴) ⊆ [𝐴]𝑟))
65adantr 481 . . . . . . 7 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 → ran (𝑇𝐴) ⊆ [𝐴]𝑟))
7 ssel 3577 . . . . . . . . 9 (ran (𝑇𝐴) ⊆ [𝐴]𝑟 → (𝐵 ∈ ran (𝑇𝐴) → 𝐵 ∈ [𝐴]𝑟))
87com12 32 . . . . . . . 8 (𝐵 ∈ ran (𝑇𝐴) → (ran (𝑇𝐴) ⊆ [𝐴]𝑟𝐵 ∈ [𝐴]𝑟))
9 simpl 473 . . . . . . . . . . 11 ((𝐵 ∈ [𝐴]𝑟𝐴𝑊) → 𝐵 ∈ [𝐴]𝑟)
10 elecg 7730 . . . . . . . . . . 11 ((𝐵 ∈ [𝐴]𝑟𝐴𝑊) → (𝐵 ∈ [𝐴]𝑟𝐴𝑟𝐵))
119, 10mpbid 222 . . . . . . . . . 10 ((𝐵 ∈ [𝐴]𝑟𝐴𝑊) → 𝐴𝑟𝐵)
12 df-br 4614 . . . . . . . . . 10 (𝐴𝑟𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑟)
1311, 12sylib 208 . . . . . . . . 9 ((𝐵 ∈ [𝐴]𝑟𝐴𝑊) → ⟨𝐴, 𝐵⟩ ∈ 𝑟)
1413expcom 451 . . . . . . . 8 (𝐴𝑊 → (𝐵 ∈ [𝐴]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
158, 14sylan9r 689 . . . . . . 7 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → (ran (𝑇𝐴) ⊆ [𝐴]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
166, 15syld 47 . . . . . 6 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
1716adantld 483 . . . . 5 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → ((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
1817alrimiv 1852 . . . 4 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
19 opex 4893 . . . . 5 𝐴, 𝐵⟩ ∈ V
2019elintab 4452 . . . 4 (⟨𝐴, 𝐵⟩ ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ↔ ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
2118, 20sylibr 224 . . 3 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → ⟨𝐴, 𝐵⟩ ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)})
22 efgval.w . . . 4 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
23 efgval.r . . . 4 = ( ~FG𝐼)
24 efgval2.m . . . 4 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
25 efgval2.t . . . 4 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
2622, 23, 24, 25efgval2 18058 . . 3 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)}
2721, 26syl6eleqr 2709 . 2 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → ⟨𝐴, 𝐵⟩ ∈ )
28 df-br 4614 . 2 (𝐴 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ )
2927, 28sylibr 224 1 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2907  cdif 3552  wss 3555  cop 4154  cotp 4156   cint 4440   class class class wbr 4613  cmpt 4673   I cid 4984   × cxp 5072  ran crn 5075  cfv 5847  (class class class)co 6604  cmpt2 6606  1𝑜c1o 7498  2𝑜c2o 7499   Er wer 7684  [cec 7685  0cc0 9880  ...cfz 12268  #chash 13057  Word cword 13230   splice csplice 13235  ⟨“cs2 13523   ~FG cefg 18040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-ec 7689  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-s2 13530  df-efg 18043
This theorem is referenced by:  efginvrel2  18061  efgsrel  18068  efgcpbllemb  18089
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