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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 26869 | . 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2) |
4 | 2re 11705 | . . . . . 6 ⊢ 2 ∈ ℝ | |
5 | ltpnf 12509 | . . . . . 6 ⊢ (2 ∈ ℝ → 2 < +∞) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 2 < +∞ |
7 | 4 | rexri 10693 | . . . . . 6 ⊢ 2 ∈ ℝ* |
8 | pnfxr 10689 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
9 | xrltnle 10702 | . . . . . 6 ⊢ ((2 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (2 < +∞ ↔ ¬ +∞ ≤ 2)) | |
10 | 7, 8, 9 | mp2an 690 | . . . . 5 ⊢ (2 < +∞ ↔ ¬ +∞ ≤ 2) |
11 | 6, 10 | mpbi 232 | . . . 4 ⊢ ¬ +∞ ≤ 2 |
12 | fvex 6677 | . . . . . 6 ⊢ (𝐸‘𝐹) ∈ V | |
13 | hashinf 13689 | . . . . . 6 ⊢ (((𝐸‘𝐹) ∈ V ∧ ¬ (𝐸‘𝐹) ∈ Fin) → (♯‘(𝐸‘𝐹)) = +∞) | |
14 | 12, 13 | mpan 688 | . . . . 5 ⊢ (¬ (𝐸‘𝐹) ∈ Fin → (♯‘(𝐸‘𝐹)) = +∞) |
15 | 14 | breq1d 5068 | . . . 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 1533 ∈ wcel 2110 Vcvv 3494 class class class wbr 5058 Fn wfn 6344 ‘cfv 6349 Fincfn 8503 ℝcr 10530 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 2c2 11686 ♯chash 13684 Vtxcvtx 26775 iEdgciedg 26776 UPGraphcupgr 26859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-hash 13685 df-upgr 26861 |
This theorem is referenced by: upgrex 26871 |
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