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Mirrors > Home > MPE Home > Th. List > upgrfi | Structured version Visualization version GIF version |
Description: An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.) |
Ref | Expression |
---|---|
isupgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
isupgr.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
upgrfi | ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isupgr.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | isupgr.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | upgrle 26438 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2) |
4 | 2re 11449 | . . . . . 6 ⊢ 2 ∈ ℝ | |
5 | ltpnf 12265 | . . . . . 6 ⊢ (2 ∈ ℝ → 2 < +∞) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 2 < +∞ |
7 | 4 | rexri 10435 | . . . . . 6 ⊢ 2 ∈ ℝ* |
8 | pnfxr 10430 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
9 | xrltnle 10444 | . . . . . 6 ⊢ ((2 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (2 < +∞ ↔ ¬ +∞ ≤ 2)) | |
10 | 7, 8, 9 | mp2an 682 | . . . . 5 ⊢ (2 < +∞ ↔ ¬ +∞ ≤ 2) |
11 | 6, 10 | mpbi 222 | . . . 4 ⊢ ¬ +∞ ≤ 2 |
12 | fvex 6459 | . . . . . 6 ⊢ (𝐸‘𝐹) ∈ V | |
13 | hashinf 13440 | . . . . . 6 ⊢ (((𝐸‘𝐹) ∈ V ∧ ¬ (𝐸‘𝐹) ∈ Fin) → (♯‘(𝐸‘𝐹)) = +∞) | |
14 | 12, 13 | mpan 680 | . . . . 5 ⊢ (¬ (𝐸‘𝐹) ∈ Fin → (♯‘(𝐸‘𝐹)) = +∞) |
15 | 14 | breq1d 4896 | . . . 4 ⊢ (¬ (𝐸‘𝐹) ∈ Fin → ((♯‘(𝐸‘𝐹)) ≤ 2 ↔ +∞ ≤ 2)) |
16 | 11, 15 | mtbiri 319 | . . 3 ⊢ (¬ (𝐸‘𝐹) ∈ Fin → ¬ (♯‘(𝐸‘𝐹)) ≤ 2) |
17 | 16 | con4i 114 | . 2 ⊢ ((♯‘(𝐸‘𝐹)) ≤ 2 → (𝐸‘𝐹) ∈ Fin) |
18 | 3, 17 | syl 17 | 1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 Vcvv 3398 class class class wbr 4886 Fn wfn 6130 ‘cfv 6135 Fincfn 8241 ℝcr 10271 +∞cpnf 10408 ℝ*cxr 10410 < clt 10411 ≤ cle 10412 2c2 11430 ♯chash 13435 Vtxcvtx 26344 iEdgciedg 26345 UPGraphcupgr 26428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-n0 11643 df-z 11729 df-uz 11993 df-hash 13436 df-upgr 26430 |
This theorem is referenced by: upgrex 26440 |
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