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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 14551) 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 4875 | . . . . . . 7 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → 〈𝑎, 𝑏〉 = 〈𝑉, 𝐸〉) | |
| 4 | 3 | eleq1d 2826 | . . . . . 6 ⊢ ((𝑎 = 𝑉 ∧ 𝑏 = 𝐸) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
| 5 | 2, 4 | sbcied 3832 | . . . . 5 ⊢ (𝑎 = 𝑉 → ([𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑉, 𝐸〉 ∈ 𝐺)) |
| 6 | 5 | sbcieg 3828 | . . . 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 1133 | . 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 3484 | . . . . . 6 ⊢ 𝑣 ∈ V | |
| 14 | vex 3484 | . . . . . 6 ⊢ 𝑒 ∈ V | |
| 15 | opeq12 4875 | . . . . . . 7 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → 〈𝑎, 𝑏〉 = 〈𝑣, 𝑒〉) | |
| 16 | 15 | eleq1d 2826 | . . . . . 6 ⊢ ((𝑎 = 𝑣 ∧ 𝑏 = 𝑒) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈𝑣, 𝑒〉 ∈ 𝐺)) |
| 17 | 13, 14, 16 | sbc2ie 3866 | . . . . 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 5330 | . . . . 5 ⊢ (𝑣 ∖ {𝑛}) ∈ V |
| 21 | opeq12 4875 | . . . . . 6 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → 〈𝑎, 𝑏〉 = 〈(𝑣 ∖ {𝑛}), 𝐹〉) | |
| 22 | 21 | eleq1d 2826 | . . . . 5 ⊢ ((𝑎 = (𝑣 ∖ {𝑛}) ∧ 𝑏 = 𝐹) → (〈𝑎, 𝑏〉 ∈ 𝐺 ↔ 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺)) |
| 23 | 20, 9, 22 | sbc2ie 3866 | . . . 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 1158 | . . . . 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 14546 | . 2 ⊢ (([𝑉 / 𝑎][𝐸 / 𝑏]〈𝑎, 𝑏〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑) |
| 33 | 8, 32 | syld3an1 1412 | 1 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 𝐿 ≤ (♯‘𝑉)) → 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 [wsbc 3788 ∖ cdif 3948 {csn 4626 〈cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 1c1 11156 + caddc 11158 ≤ cle 11296 ℕ0cn0 12526 ♯chash 14369 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-hash 14370 |
| This theorem is referenced by: opfi1ind 14551 |
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