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Theorem brfi1uzind 14312
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 14313) 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
21brrelex12i 5673 . . 3 (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3 simpl 483 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
4 simplr 766 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → 𝐸 ∈ V)
5 breq12 5097 . . . . . . 7 ((𝑎 = 𝑉𝑏 = 𝐸) → (𝑎𝐺𝑏𝑉𝐺𝐸))
65adantll 711 . . . . . 6 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) ∧ 𝑏 = 𝐸) → (𝑎𝐺𝑏𝑉𝐺𝐸))
74, 6sbcied 3772 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → ([𝐸 / 𝑏]𝑎𝐺𝑏𝑉𝐺𝐸))
83, 7sbcied 3772 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → ([𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏𝑉𝐺𝐸))
98biimprcd 249 . . 3 (𝑉𝐺𝐸 → ((𝑉 ∈ V ∧ 𝐸 ∈ V) → [𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏))
102, 9mpd 15 . 2 (𝑉𝐺𝐸[𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏)
11 brfi1uzind.f . . 3 𝐹 ∈ V
12 brfi1uzind.l . . 3 𝐿 ∈ ℕ0
13 brfi1uzind.1 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
14 brfi1uzind.2 . . 3 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
15 vex 3445 . . . . 5 𝑣 ∈ V
16 vex 3445 . . . . 5 𝑒 ∈ V
17 breq12 5097 . . . . 5 ((𝑎 = 𝑣𝑏 = 𝑒) → (𝑎𝐺𝑏𝑣𝐺𝑒))
1815, 16, 17sbc2ie 3810 . . . 4 ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏𝑣𝐺𝑒)
19 brfi1uzind.3 . . . . 5 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
2015difexi 5272 . . . . . 6 (𝑣 ∖ {𝑛}) ∈ V
21 breq12 5097 . . . . . 6 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
2220, 11, 21sbc2ie 3810 . . . . 5 ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹)
2319, 22sylibr 233 . . . 4 ((𝑣𝐺𝑒𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏)
2418, 23sylanb 581 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏)
25 brfi1uzind.4 . . 3 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
26 brfi1uzind.base . . . 4 ((𝑣𝐺𝑒 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
2718, 26sylanb 581 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
28183anbi1i 1156 . . . . 5 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
2928anbi2i 623 . . . 4 (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)))
30 brfi1uzind.step . . . 4 ((((𝑦 + 1) ∈ ℕ0 ∧ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
3129, 30sylanb 581 . . 3 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
3211, 12, 13, 14, 24, 25, 27, 31fi1uzind 14311 . 2 (([𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
3310, 32syl3an1 1162 1 ((𝑉𝐺𝐸𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  Vcvv 3441  [wsbc 3727  cdif 3895  {csn 4573   class class class wbr 5092  Rel wrel 5625  cfv 6479  (class class class)co 7337  Fincfn 8804  1c1 10973   + caddc 10975  cle 11111  0cn0 12334  chash 14145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-oadd 8371  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-dju 9758  df-card 9796  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-n0 12335  df-xnn0 12407  df-z 12421  df-uz 12684  df-fz 13341  df-hash 14146
This theorem is referenced by:  brfi1ind  14313
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