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Theorem brfi1uzind 13135
 Description: Properties of a binary relation with a finite first component with at least L elements, proven by finite induction on the size of the first component. This theorem can be applied for graphs (as binary relation between the set of vertices and an edge function) with a finite number of vertices, usually with 𝐿 = 0 (see brfi1ind 13136) or 𝐿 = 1. (Contributed by Alexander van der Vekens, 7-Jan-2018.) (Proof shortened by AV, 23-Oct-2020.) (Revised by AV, 28-Mar-2021.)
Hypotheses
Ref Expression
brfi1uzind.r Rel 𝐺
brfi1uzind.f 𝐹 ∈ V
brfi1uzind.l 𝐿 ∈ ℕ0
brfi1uzind.1 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
brfi1uzind.2 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
brfi1uzind.3 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
brfi1uzind.4 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
brfi1uzind.base ((𝑣𝐺𝑒 ∧ (#‘𝑣) = 𝐿) → 𝜓)
brfi1uzind.step ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (#‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
Assertion
Ref Expression
brfi1uzind ((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)   𝐸(𝑦,𝑤,𝑓)   𝐹(𝑦,𝑣,𝑒,𝑛)   𝐿(𝑤,𝑓)   𝑉(𝑦,𝑤,𝑓)

Proof of Theorem brfi1uzind
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brfi1uzind.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 brfi1uzind.f . . 3 𝐹 ∈ V
13 brfi1uzind.l . . 3 𝐿 ∈ ℕ0
14 brfi1uzind.1 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
15 brfi1uzind.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 brfi1uzind.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 brfi1uzind.4 . . 3 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
29 brfi1uzind.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 brfi1uzind.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, 34fi1uzind 13134 . 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:  brfi1ind  13136
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