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Mirrors > Home > MPE Home > Th. List > opfi1uzind | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
opfi1uzind.e | ⊢ 𝐸 ∈ V |
opfi1uzind.f | ⊢ 𝐹 ∈ V |
opfi1uzind.l | ⊢ 𝐿 ∈ ℕ0 |
opfi1uzind.1 | ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑)) |
opfi1uzind.2 | ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃)) |
opfi1uzind.3 | ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
opfi1uzind.4 | ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒)) |
opfi1uzind.base | ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓) |
opfi1uzind.step | ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓) |
Ref | Expression |
---|---|
opfi1uzind | ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opfi1uzind.e | . . . . . . 7 ⊢ 𝐸 ∈ V | |
2 | 1 | a1i 11 | . . . . . 6 ⊢ (𝑎 = 𝑉 → 𝐸 ∈ V) |
3 | opeq12 4627 | . . . . . . 7 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → 〈𝑎, 𝑏〉 = 〈𝑉, 𝐸〉) | |
4 | 3 | eleq1d 2891 | . . . . . 6 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
5 | 2, 4 | sbcied 3699 | . . . . 5 ⊢ (𝑎 = 𝑉 → ([𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
6 | 5 | sbcieg 3695 | . . . 4 ⊢ (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
7 | 6 | biimparc 473 | . . 3 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin) → [𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺) |
8 | 7 | 3adant3 1166 | . 2 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ 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 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑)) | |
12 | opfi1uzind.2 | . . 3 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃)) | |
13 | vex 3417 | . . . . . 6 ⊢ 𝑣 ∈ V | |
14 | vex 3417 | . . . . . 6 ⊢ 𝑒 ∈ V | |
15 | opeq12 4627 | . . . . . . 7 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → 〈𝑎, 𝑏〉 = 〈𝑣, 𝑒〉) | |
16 | 15 | eleq1d 2891 | . . . . . 6 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑣, 𝑒〉 ∈ 𝐺)) |
17 | 13, 14, 16 | sbc2ie 3730 | . . . . 5 ⊢ ([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑣, 𝑒〉 ∈ 𝐺) |
18 | opfi1uzind.3 | . . . . 5 ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) | |
19 | 17, 18 | sylanb 576 | . . . 4 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
20 | 13 | difexi 5036 | . . . . 5 ⊢ (𝑣 ∖ {𝑛}) ∈ V |
21 | opeq12 4627 | . . . . . 6 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → 〈𝑎, 𝑏〉 = 〈(𝑣 ∖ {𝑛}), 𝐹〉) | |
22 | 21 | eleq1d 2891 | . . . . 5 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺)) |
23 | 20, 9, 22 | sbc2ie 3730 | . . . 4 ⊢ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
24 | 19, 23 | sylibr 226 | . . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → [(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺) |
25 | opfi1uzind.4 | . . 3 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒)) | |
26 | opfi1uzind.base | . . . 4 ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓) | |
27 | 17, 26 | sylanb 576 | . . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓) |
28 | 17 | 3anbi1i 1200 | . . . . 5 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) ↔ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) |
29 | 28 | anbi2i 616 | . . . 4 ⊢ (((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ↔ ((𝑦 + 1) ∈ ℕ0 ∧ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣))) |
30 | opfi1uzind.step | . . . 4 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓) | |
31 | 29, 30 | sylanb 576 | . . 3 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ ([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓) |
32 | 9, 10, 11, 12, 24, 25, 27, 31 | fi1uzind 13575 | . 2 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑) |
33 | 8, 32 | syld3an1 1533 | 1 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ 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 ℕ0cn0 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 |
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