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Theorem brfi1uzind 14489
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 14490) 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 5727 . . 3 (𝑉𝐺𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
3 simpl 481 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
4 simplr 767 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → 𝐸 ∈ V)
5 breq12 5148 . . . . . . 7 ((𝑎 = 𝑉𝑏 = 𝐸) → (𝑎𝐺𝑏𝑉𝐺𝐸))
65adantll 712 . . . . . 6 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) ∧ 𝑏 = 𝐸) → (𝑎𝐺𝑏𝑉𝐺𝐸))
74, 6sbcied 3815 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑎 = 𝑉) → ([𝐸 / 𝑏]𝑎𝐺𝑏𝑉𝐺𝐸))
83, 7sbcied 3815 . . . 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 3467 . . . . 5 𝑣 ∈ V
16 vex 3467 . . . . 5 𝑒 ∈ V
17 breq12 5148 . . . . 5 ((𝑎 = 𝑣𝑏 = 𝑒) → (𝑎𝐺𝑏𝑣𝐺𝑒))
1815, 16, 17sbc2ie 3852 . . . 4 ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏𝑣𝐺𝑒)
19 brfi1uzind.3 . . . . 5 ((𝑣𝐺𝑒𝑛𝑣) → (𝑣 ∖ {𝑛})𝐺𝐹)
2015difexi 5325 . . . . . 6 (𝑣 ∖ {𝑛}) ∈ V
21 breq12 5148 . . . . . 6 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹))
2220, 11, 21sbc2ie 3852 . . . . 5 ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏 ↔ (𝑣 ∖ {𝑛})𝐺𝐹)
2319, 22sylibr 233 . . . 4 ((𝑣𝐺𝑒𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎𝐺𝑏)
2418, 23sylanb 579 . . 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 579 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
28183anbi1i 1154 . . . . 5 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) ↔ (𝑣𝐺𝑒 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
2928anbi2i 621 . . . 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 579 . . 3 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎𝐺𝑏 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
3211, 12, 13, 14, 24, 25, 27, 31fi1uzind 14488 . 2 (([𝑉 / 𝑎][𝐸 / 𝑏]𝑎𝐺𝑏𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
3310, 32syl3an1 1160 1 ((𝑉𝐺𝐸𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3463  [wsbc 3769  cdif 3937  {csn 4624   class class class wbr 5143  Rel wrel 5677  cfv 6542  (class class class)co 7415  Fincfn 8960  1c1 11137   + caddc 11139  cle 11277  0cn0 12500  chash 14319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-oadd 8487  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-dju 9922  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-n0 12501  df-xnn0 12573  df-z 12587  df-uz 12851  df-fz 13515  df-hash 14320
This theorem is referenced by:  brfi1ind  14490
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