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Mirrors > Home > MPE Home > Th. List > fusgrfisbase | Structured version Visualization version GIF version |
Description: Induction base for fusgrfis 27449. Main work is done in uhgr0v0e 27357. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.) |
Ref | Expression |
---|---|
fusgrfisbase | ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → 𝐸 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgruhgr 27305 | . . . . 5 ⊢ (〈𝑉, 𝐸〉 ∈ USGraph → 〈𝑉, 𝐸〉 ∈ UHGraph) | |
2 | 1 | 3ad2ant2 1136 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → 〈𝑉, 𝐸〉 ∈ UHGraph) |
3 | opvtxfv 27126 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
4 | 3 | 3ad2ant1 1135 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) |
5 | hasheq0 13962 | . . . . . . . . 9 ⊢ (𝑉 ∈ 𝑋 → ((♯‘𝑉) = 0 ↔ 𝑉 = ∅)) | |
6 | 5 | biimpd 232 | . . . . . . . 8 ⊢ (𝑉 ∈ 𝑋 → ((♯‘𝑉) = 0 → 𝑉 = ∅)) |
7 | 6 | adantr 484 | . . . . . . 7 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((♯‘𝑉) = 0 → 𝑉 = ∅)) |
8 | 7 | a1d 25 | . . . . . 6 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (〈𝑉, 𝐸〉 ∈ USGraph → ((♯‘𝑉) = 0 → 𝑉 = ∅))) |
9 | 8 | 3imp 1113 | . . . . 5 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → 𝑉 = ∅) |
10 | 4, 9 | eqtrd 2779 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → (Vtx‘〈𝑉, 𝐸〉) = ∅) |
11 | eqid 2739 | . . . . 5 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
12 | eqid 2739 | . . . . 5 ⊢ (Edg‘〈𝑉, 𝐸〉) = (Edg‘〈𝑉, 𝐸〉) | |
13 | 11, 12 | uhgr0v0e 27357 | . . . 4 ⊢ ((〈𝑉, 𝐸〉 ∈ UHGraph ∧ (Vtx‘〈𝑉, 𝐸〉) = ∅) → (Edg‘〈𝑉, 𝐸〉) = ∅) |
14 | 2, 10, 13 | syl2anc 587 | . . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → (Edg‘〈𝑉, 𝐸〉) = ∅) |
15 | 0fin 8874 | . . 3 ⊢ ∅ ∈ Fin | |
16 | 14, 15 | eqeltrdi 2848 | . 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → (Edg‘〈𝑉, 𝐸〉) ∈ Fin) |
17 | eqid 2739 | . . . . 5 ⊢ (iEdg‘〈𝑉, 𝐸〉) = (iEdg‘〈𝑉, 𝐸〉) | |
18 | 17, 12 | usgredgffibi 27443 | . . . 4 ⊢ (〈𝑉, 𝐸〉 ∈ USGraph → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ (iEdg‘〈𝑉, 𝐸〉) ∈ Fin)) |
19 | 18 | 3ad2ant2 1136 | . . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ (iEdg‘〈𝑉, 𝐸〉) ∈ Fin)) |
20 | opiedgfv 27129 | . . . . 5 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) | |
21 | 20 | 3ad2ant1 1135 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) |
22 | 21 | eleq1d 2824 | . . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → ((iEdg‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝐸 ∈ Fin)) |
23 | 19, 22 | bitrd 282 | . 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → ((Edg‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝐸 ∈ Fin)) |
24 | 16, 23 | mpbid 235 | 1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 〈𝑉, 𝐸〉 ∈ USGraph ∧ (♯‘𝑉) = 0) → 𝐸 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∅c0 4253 〈cop 4563 ‘cfv 6400 Fincfn 8649 0cc0 10758 ♯chash 13928 Vtxcvtx 27118 iEdgciedg 27119 Edgcedg 27169 UHGraphcuhgr 27178 USGraphcusgr 27271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5195 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10814 ax-resscn 10815 ax-1cn 10816 ax-icn 10817 ax-addcl 10818 ax-addrcl 10819 ax-mulcl 10820 ax-mulrcl 10821 ax-mulcom 10822 ax-addass 10823 ax-mulass 10824 ax-distr 10825 ax-i2m1 10826 ax-1ne0 10827 ax-1rid 10828 ax-rnegex 10829 ax-rrecex 10830 ax-cnre 10831 ax-pre-lttri 10832 ax-pre-lttrn 10833 ax-pre-ltadd 10834 ax-pre-mulgt0 10835 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-int 4876 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-1st 7782 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-1o 8225 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-fin 8653 df-card 9584 df-pnf 10898 df-mnf 10899 df-xr 10900 df-ltxr 10901 df-le 10902 df-sub 11093 df-neg 11094 df-nn 11860 df-2 11922 df-n0 12120 df-z 12206 df-uz 12468 df-fz 13125 df-hash 13929 df-vtx 27120 df-iedg 27121 df-edg 27170 df-uhgr 27180 df-upgr 27204 df-uspgr 27272 df-usgr 27273 |
This theorem is referenced by: fusgrfis 27449 |
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