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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 13863) 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 4807 | . . . . . . 7 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → 〈𝑎, 𝑏〉 = 〈𝑉, 𝐸〉) | |
4 | 3 | eleq1d 2899 | . . . . . 6 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
5 | 2, 4 | sbcied 3816 | . . . . 5 ⊢ (𝑎 = 𝑉 → ([𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
6 | 5 | sbcieg 3812 | . . . 4 ⊢ (𝑉 ∈ Fin → ([𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
7 | 6 | biimparc 482 | . . 3 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin) → [𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺) |
8 | 7 | 3adant3 1128 | . 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 3499 | . . . . . 6 ⊢ 𝑣 ∈ V | |
14 | vex 3499 | . . . . . 6 ⊢ 𝑒 ∈ V | |
15 | opeq12 4807 | . . . . . . 7 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → 〈𝑎, 𝑏〉 = 〈𝑣, 𝑒〉) | |
16 | 15 | eleq1d 2899 | . . . . . 6 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑣, 𝑒〉 ∈ 𝐺)) |
17 | 13, 14, 16 | sbc2ie 3852 | . . . . 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 583 | . . . 4 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
20 | 13 | difexi 5234 | . . . . 5 ⊢ (𝑣 ∖ {𝑛}) ∈ V |
21 | opeq12 4807 | . . . . . 6 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → 〈𝑎, 𝑏〉 = 〈(𝑣 ∖ {𝑛}), 𝐹〉) | |
22 | 21 | eleq1d 2899 | . . . . 5 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺)) |
23 | 20, 9, 22 | sbc2ie 3852 | . . . 4 ⊢ ([(𝑣 ∖ {𝑛}) / 𝑎][𝐹 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
24 | 19, 23 | sylibr 236 | . . 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 583 | . . 3 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (♯‘𝑣) = 𝐿) → 𝜓) |
28 | 17 | 3anbi1i 1153 | . . . . 5 ⊢ (([𝑣 / 𝑎][𝑒 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣) ↔ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) |
29 | 28 | anbi2i 624 | . . . 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 583 | . . 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 13858 | . 2 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑) |
33 | 8, 32 | syld3an1 1406 | 1 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3496 [wsbc 3774 ∖ cdif 3935 {csn 4569 〈cop 4575 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 1c1 10540 + caddc 10542 ≤ cle 10678 ℕ0cn0 11900 ♯chash 13693 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: opfi1ind 13863 |
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