MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opfi1uzind Structured version   Visualization version   GIF version

Theorem opfi1uzind 13579
Description: Properties of an ordered pair 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 (represented as orderd pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 13580) or 𝐿 = 1. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.)
Hypotheses
Ref Expression
opfi1uzind.e 𝐸 ∈ V
opfi1uzind.f 𝐹 ∈ V
opfi1uzind.l 𝐿 ∈ ℕ0
opfi1uzind.1 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
opfi1uzind.2 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
opfi1uzind.3 ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
opfi1uzind.4 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
opfi1uzind.base ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
opfi1uzind.step ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
Assertion
Ref Expression
opfi1uzind ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
Distinct variable groups:   𝑒,𝑛,𝑣,𝑦   𝑒,𝐸,𝑛,𝑣   𝑓,𝐹,𝑤   𝑒,𝐺,𝑓,𝑛,𝑣,𝑤,𝑦   𝑒,𝑉,𝑛,𝑣   𝜓,𝑓,𝑛,𝑤,𝑦   𝜃,𝑒,𝑛,𝑣   𝜒,𝑓,𝑤   𝜑,𝑒,𝑛,𝑣   𝑒,𝐿,𝑛,𝑣,𝑦
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 opfi1uzind
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opfi1uzind.e . . . . . . 7 𝐸 ∈ V
21a1i 11 . . . . . 6 (𝑎 = 𝑉𝐸 ∈ V)
3 opeq12 4627 . . . . . . 7 ((𝑎 = 𝑉𝑏 = 𝐸) → ⟨𝑎, 𝑏⟩ = ⟨𝑉, 𝐸⟩)
43eleq1d 2891 . . . . . 6 ((𝑎 = 𝑉𝑏 = 𝐸) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
52, 4sbcied 3699 . . . . 5 (𝑎 = 𝑉 → ([𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
65sbcieg 3695 . . . 4 (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑉, 𝐸⟩ ∈ 𝐺))
76biimparc 473 . . 3 ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin) → [𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺)
873adant3 1166 . 2 ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → [𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺)
9 opfi1uzind.f . . 3 𝐹 ∈ V
10 opfi1uzind.l . . 3 𝐿 ∈ ℕ0
11 opfi1uzind.1 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝜓𝜑))
12 opfi1uzind.2 . . 3 ((𝑣 = 𝑤𝑒 = 𝑓) → (𝜓𝜃))
13 vex 3417 . . . . . 6 𝑣 ∈ V
14 vex 3417 . . . . . 6 𝑒 ∈ V
15 opeq12 4627 . . . . . . 7 ((𝑎 = 𝑣𝑏 = 𝑒) → ⟨𝑎, 𝑏⟩ = ⟨𝑣, 𝑒⟩)
1615eleq1d 2891 . . . . . 6 ((𝑎 = 𝑣𝑏 = 𝑒) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑣, 𝑒⟩ ∈ 𝐺))
1713, 14, 16sbc2ie 3730 . . . . 5 ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨𝑣, 𝑒⟩ ∈ 𝐺)
18 opfi1uzind.3 . . . . 5 ((⟨𝑣, 𝑒⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
1917, 18sylanb 576 . . . 4 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺𝑛𝑣) → ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
2013difexi 5036 . . . . 5 (𝑣 ∖ {𝑛}) ∈ V
21 opeq12 4627 . . . . . 6 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → ⟨𝑎, 𝑏⟩ = ⟨(𝑣 ∖ {𝑛}), 𝐹⟩)
2221eleq1d 2891 . . . . 5 ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (⟨𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺))
2320, 9, 22sbc2ie 3730 . . . 4 ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ↔ ⟨(𝑣 ∖ {𝑛}), 𝐹⟩ ∈ 𝐺)
2419, 23sylibr 226 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺𝑛𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺)
25 opfi1uzind.4 . . 3 ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃𝜒))
26 opfi1uzind.base . . . 4 ((⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
2717, 26sylanb 576 . . 3 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓)
28173anbi1i 1200 . . . . 5 (([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣) ↔ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣))
2928anbi2i 616 . . . 4 (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)))
30 opfi1uzind.step . . . 4 ((((𝑦 + 1) ∈ ℕ0 ∧ (⟨𝑣, 𝑒⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
3129, 30sylanb 576 . . 3 ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛𝑣)) ∧ 𝜒) → 𝜓)
329, 10, 11, 12, 24, 25, 27, 31fi1uzind 13575 . 2 (([𝑉 / 𝑎][𝐸 / 𝑏]𝑎, 𝑏⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
338, 32syld3an1 1533 1 ((⟨𝑉, 𝐸⟩ ∈ 𝐺𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1111   = wceq 1656  wcel 2164  Vcvv 3414  [wsbc 3662  cdif 3795  {csn 4399  cop 4405   class class class wbr 4875  cfv 6127  (class class class)co 6910  Fincfn 8228  1c1 10260   + caddc 10262  cle 10399  0cn0 11625  chash 13417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-oadd 7835  df-er 8014  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-fz 12627  df-hash 13418
This theorem is referenced by:  opfi1ind  13580
  Copyright terms: Public domain W3C validator