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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 ordered pairs of vertices and edges) with a finite number of vertices, usually with 𝐿 = 0 (see opfi1ind 14547) 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 4879 | . . . . . . 7 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → 〈𝑎, 𝑏〉 = 〈𝑉, 𝐸〉) | |
4 | 3 | eleq1d 2823 | . . . . . 6 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
5 | 2, 4 | sbcied 3836 | . . . . 5 ⊢ (𝑎 = 𝑉 → ([𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
6 | 5 | sbcieg 3831 | . . . 4 ⊢ (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
7 | 6 | biimparc 479 | . . 3 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin) → [𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺) |
8 | 7 | 3adant3 1131 | . 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 3481 | . . . . . 6 ⊢ 𝑣 ∈ V | |
14 | vex 3481 | . . . . . 6 ⊢ 𝑒 ∈ V | |
15 | opeq12 4879 | . . . . . . 7 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → 〈𝑎, 𝑏〉 = 〈𝑣, 𝑒〉) | |
16 | 15 | eleq1d 2823 | . . . . . 6 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑣, 𝑒〉 ∈ 𝐺)) |
17 | 13, 14, 16 | sbc2ie 3873 | . . . . 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 581 | . . . 4 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
20 | 13 | difexi 5335 | . . . . 5 ⊢ (𝑣 ∖ {𝑛}) ∈ V |
21 | opeq12 4879 | . . . . . 6 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → 〈𝑎, 𝑏〉 = 〈(𝑣 ∖ {𝑛}), 𝐹〉) | |
22 | 21 | eleq1d 2823 | . . . . 5 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺)) |
23 | 20, 9, 22 | sbc2ie 3873 | . . . 4 ⊢ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
24 | 19, 23 | sylibr 234 | . . 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 581 | . . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓) |
28 | 17 | 3anbi1i 1156 | . . . . 5 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) ↔ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) |
29 | 28 | anbi2i 623 | . . . 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 581 | . . 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 14542 | . 2 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑) |
33 | 8, 32 | syld3an1 1409 | 1 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 Vcvv 3477 [wsbc 3790 ∖ cdif 3959 {csn 4630 〈cop 4636 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 1c1 11153 + caddc 11155 ≤ cle 11293 ℕ0cn0 12523 ♯chash 14365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12876 df-fz 13544 df-hash 14366 |
This theorem is referenced by: opfi1ind 14547 |
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