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Theorem brfi1uzindOLD 13141
 Description: Obsolete version of brfi1uzind 13135 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 5079 . . . 4 ((Rel 𝐺𝑉𝐺𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V))
31, 2mpan 702 . . 3 (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
4 simpl 472 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
5 simplr 788 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → 𝐸 ∈ V)
6 breq12 4588 . . . . . . 7 ((𝑎 = 𝑉𝑏 = 𝐸) → (𝑎𝐺𝑏𝑉𝐺𝐸))
76adantll 746 . . . . . 6 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) ∧ 𝑏 = 𝐸) → (𝑎𝐺𝑏𝑉𝐺𝐸))
85, 7sbcied 3439 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → ([𝐸 / 𝑏]𝑎𝐺𝑏𝑉𝐺𝐸))
94, 8sbcied 3439 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏𝑉𝐺𝐸))
109biimprcd 239 . . 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 3176 . . . . 5 𝑣 ∈ V
17 vex 3176 . . . . 5 𝑒 ∈ V
18 breq12 4588 . . . . 5 ((𝑎 = 𝑣𝑏 = 𝑒) → (𝑎𝐺𝑏𝑣𝐺𝑒))
1916, 17, 18sbc2ie 3472 . . . 4 ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏𝑣𝐺𝑒)
20 brfi1uzindOLD.3 . . . . 5 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
21 difexg 4735 . . . . . . 7 (𝑣 ∈ V → (𝑣 ∖ {𝑛}) ∈ V)
2216, 21ax-mp 5 . . . . . 6 (𝑣 ∖ {𝑛}) ∈ V
2312elexi 3186 . . . . . 6 𝐹 ∈ V
24 breq12 4588 . . . . . 6 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
2522, 23, 24sbc2ie 3472 . . . . 5 ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹)
2620, 25sylibr 223 . . . 4 ((𝑣𝐺𝑒𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏)
2719, 26sylanb 488 . . 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 488 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (#‘𝑣) = 𝐿) → 𝜓)
31193anbi1i 1246 . . . . 5 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) ↔ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
3231anbi2i 726 . . . 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 488 . . 3 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
3512, 13, 14, 15, 27, 28, 30, 34fi1uzindOLD 13140 . 2 (([𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
3611, 35syl3an1 1351 1 ((𝑉𝐺𝐸𝑉 ∈ Fin ∧ 𝐿 ≤ (#‘𝑉)) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402   ∖ cdif 3537  {csn 4125   class class class wbr 4583  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  1c1 9816   + caddc 9818   ≤ cle 9954  ℕ0cn0 11169  #chash 12979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980 This theorem is referenced by:  brfi1indOLD  13142
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