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Theorem brfi1uzindOLD 13225
Description: Obsolete version of brfi1uzind 13219 as of 28-Mar-2021. (Contributed by AV, 7-Jan-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
brfi1uzindOLD.r Rel 𝐺
brfi1uzindOLD.f 𝐹𝑈
brfi1uzindOLD.l 𝐿 ∈ ℕ0
brfi1uzindOLD.1 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
brfi1uzindOLD.2 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
brfi1uzindOLD.3 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
brfi1uzindOLD.4 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
brfi1uzindOLD.base ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝐿) → 𝜓)
brfi1uzindOLD.step ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
Assertion
Ref Expression
brfi1uzindOLD ((𝑉𝐺𝐸𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
Distinct variable groups:   𝑒,𝐸,𝑛,𝑣   𝑓,𝐹,𝑤   𝑒,𝐺,𝑓,𝑛,𝑣,𝑤,𝑦   𝑒,𝐿,𝑛,𝑣,𝑦   𝑒,𝑉,𝑛,𝑣   𝜓,𝑓,𝑛,𝑤,𝑦   𝜃,𝑒,𝑛,𝑣   𝜒,𝑓,𝑤   𝜑,𝑒,𝑛,𝑣
Allowed substitution hints:   𝜑(𝑦,𝑤,𝑓)   𝜓(𝑣,𝑒)   𝜒(𝑦,𝑣,𝑒,𝑛)   𝜃(𝑦,𝑤,𝑓)   𝑈(𝑦,𝑤,𝑣,𝑒,𝑓,𝑛)   𝐸(𝑦,𝑤,𝑓)   𝐹(𝑦,𝑣,𝑒,𝑛)   𝐿(𝑤,𝑓)   𝑉(𝑦,𝑤,𝑓)

Proof of Theorem brfi1uzindOLD
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brfi1uzindOLD.r . . . 4 Rel 𝐺
2 brrelex12 5115 . . . 4 ((Rel 𝐺𝑉𝐺𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
31, 2mpan 705 . . 3 (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
4 simpl 473 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
5 simplr 791 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → 𝐸 ∈ V)
6 breq12 4618 . . . . . . 7 ((𝑎 = 𝑉𝑏 = 𝐸) → (𝑎𝐺𝑏𝑉𝐺𝐸))
76adantll 749 . . . . . 6 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) ∧ 𝑏 = 𝐸) → (𝑎𝐺𝑏𝑉𝐺𝐸))
85, 7sbcied 3454 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → ([𝐸 / 𝑏]𝑎𝐺𝑏𝑉𝐺𝐸))
94, 8sbcied 3454 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏𝑉𝐺𝐸))
109biimprcd 240 . . 3 (𝑉𝐺𝐸 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) → [𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏))
113, 10mpd 15 . 2 (𝑉𝐺𝐸[𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏)
12 brfi1uzindOLD.f . . 3 𝐹𝑈
13 brfi1uzindOLD.l . . 3 𝐿 ∈ ℕ0
14 brfi1uzindOLD.1 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
15 brfi1uzindOLD.2 . . 3 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
16 vex 3189 . . . . 5 𝑣 ∈ V
17 vex 3189 . . . . 5 𝑒 ∈ V
18 breq12 4618 . . . . 5 ((𝑎 = 𝑣𝑏 = 𝑒) → (𝑎𝐺𝑏𝑣𝐺𝑒))
1916, 17, 18sbc2ie 3487 . . . 4 ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏𝑣𝐺𝑒)
20 brfi1uzindOLD.3 . . . . 5 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
21 difexg 4768 . . . . . . 7 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
2216, 21ax-mp 5 . . . . . 6 (𝑣 ∖ {𝑛}) ∈ V
2312elexi 3199 . . . . . 6 𝐹 ∈ V
24 breq12 4618 . . . . . 6 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
2522, 23, 24sbc2ie 3487 . . . . 5 ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹)
2620, 25sylibr 224 . . . 4 ((𝑣𝐺𝑒𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏)
2719, 26sylanb 489 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏)
28 brfi1uzindOLD.4 . . 3 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
29 brfi1uzindOLD.base . . . 4 ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝐿) → 𝜓)
3019, 29sylanb 489 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (#‘𝑣) = 𝐿) → 𝜓)
31193anbi1i 1251 . . . . 5 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
3231anbi2i 729 . . . 4 (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)))
33 brfi1uzindOLD.step . . . 4 ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
3432, 33sylanb 489 . . 3 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
3512, 13, 14, 15, 27, 28, 30, 34fi1uzindOLD 13224 . 2 (([𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
3611, 35syl3an1 1356 1 ((𝑉𝐺𝐸𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  [wsbc 3417  cdif 3552  {csn 4148   class class class wbr 4613  Rel wrel 5079  cfv 5847  (class class class)co 6604  Fincfn 7899  1c1 9881   + caddc 9883  cle 10019  0cn0 11236  #chash 13057
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-rmo 2915  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-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-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934  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-xnn0 11308  df-z 11322  df-uz 11632  df-fz 12269  df-hash 13058
This theorem is referenced by:  brfi1indOLD  13226
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