![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 29021 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2) |
4 | 2re 12330 | . . . . . 6 ⊢ 2 ∈ ℝ | |
5 | ltpnf 13146 | . . . . . 6 ⊢ (2 ∈ ℝ → 2 < +∞) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 2 < +∞ |
7 | 4 | rexri 11311 | . . . . . 6 ⊢ 2 ∈ ℝ* |
8 | pnfxr 11307 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
9 | xrltnle 11320 | . . . . . 6 ⊢ ((2 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (2 < +∞ ↔ ¬ +∞ ≤ 2)) | |
10 | 7, 8, 9 | mp2an 690 | . . . . 5 ⊢ (2 < +∞ ↔ ¬ +∞ ≤ 2) |
11 | 6, 10 | mpbi 229 | . . . 4 ⊢ ¬ +∞ ≤ 2 |
12 | fvex 6904 | . . . . . 6 ⊢ (𝐸‘𝐹) ∈ V | |
13 | hashinf 14345 | . . . . . 6 ⊢ (((𝐸‘𝐹) ∈ V ∧ ¬ (𝐸‘𝐹) ∈ Fin) → (♯‘(𝐸‘𝐹)) = +∞) | |
14 | 12, 13 | mpan 688 | . . . . 5 ⊢ (¬ (𝐸‘𝐹) ∈ Fin → (♯‘(𝐸‘𝐹)) = +∞) |
15 | 14 | breq1d 5154 | . . . 4 ⊢ (¬ (𝐸‘𝐹) ∈ Fin → ((♯‘(𝐸‘𝐹)) ≤ 2 ↔ +∞ ≤ 2)) |
16 | 11, 15 | mtbiri 326 | . . 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 205 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 Vcvv 3463 class class class wbr 5144 Fn wfn 6539 ‘cfv 6544 Fincfn 8964 ℝcr 11146 +∞cpnf 11284 ℝ*cxr 11286 < clt 11287 ≤ cle 11288 2c2 12311 ♯chash 14340 Vtxcvtx 28927 iEdgciedg 28928 UPGraphcupgr 29011 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-n0 12517 df-z 12603 df-uz 12867 df-hash 14341 df-upgr 29013 |
This theorem is referenced by: upgrex 29023 |
Copyright terms: Public domain | W3C validator |