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 26878 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2) |
4 | 2re 11714 | . . . . . 6 ⊢ 2 ∈ ℝ | |
5 | ltpnf 12518 | . . . . . 6 ⊢ (2 ∈ ℝ → 2 < +∞) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 2 < +∞ |
7 | 4 | rexri 10702 | . . . . . 6 ⊢ 2 ∈ ℝ* |
8 | pnfxr 10698 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
9 | xrltnle 10711 | . . . . . 6 ⊢ ((2 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (2 < +∞ ↔ ¬ +∞ ≤ 2)) | |
10 | 7, 8, 9 | mp2an 690 | . . . . 5 ⊢ (2 < +∞ ↔ ¬ +∞ ≤ 2) |
11 | 6, 10 | mpbi 232 | . . . 4 ⊢ ¬ +∞ ≤ 2 |
12 | fvex 6686 | . . . . . 6 ⊢ (𝐸‘𝐹) ∈ V | |
13 | hashinf 13698 | . . . . . 6 ⊢ (((𝐸‘𝐹) ∈ V ∧ ¬ (𝐸‘𝐹) ∈ Fin) → (♯‘(𝐸‘𝐹)) = +∞) | |
14 | 12, 13 | mpan 688 | . . . . 5 ⊢ (¬ (𝐸‘𝐹) ∈ Fin → (♯‘(𝐸‘𝐹)) = +∞) |
15 | 14 | breq1d 5079 | . . . 4 ⊢ (¬ (𝐸‘𝐹) ∈ Fin → ((♯‘(𝐸‘𝐹)) ≤ 2 ↔ +∞ ≤ 2)) |
16 | 11, 15 | mtbiri 329 | . . 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 208 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 Vcvv 3497 class class class wbr 5069 Fn wfn 6353 ‘cfv 6358 Fincfn 8512 ℝcr 10539 +∞cpnf 10675 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 2c2 11695 ♯chash 13693 Vtxcvtx 26784 iEdgciedg 26785 UPGraphcupgr 26868 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-hash 13694 df-upgr 26870 |
This theorem is referenced by: upgrex 26880 |
Copyright terms: Public domain | W3C validator |