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Mirrors > Home > MPE Home > Th. List > isfusgr | Structured version Visualization version GIF version |
Description: The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
Ref | Expression |
---|---|
isfusgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
isfusgr | ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6673 | . . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
2 | isfusgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 1, 2 | syl6eqr 2877 | . . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
4 | 3 | eleq1d 2900 | . 2 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∈ Fin ↔ 𝑉 ∈ Fin)) |
5 | df-fusgr 27102 | . 2 ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin} | |
6 | 4, 5 | elrab2 3686 | 1 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 Fincfn 8512 Vtxcvtx 26784 USGraphcusgr 26937 FinUSGraphcfusgr 27101 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-fusgr 27102 |
This theorem is referenced by: fusgrvtxfi 27104 isfusgrf1 27105 isfusgrcl 27106 fusgrusgr 27107 opfusgr 27108 fusgredgfi 27110 fusgrfis 27115 cusgrsizeindslem 27236 cusgrsizeinds 27237 sizusglecusglem2 27247 fusgrmaxsize 27249 finrusgrfusgr 27350 rusgrnumwwlks 27756 rusgrnumwwlk 27757 frrusgrord0lem 28121 frrusgrord0 28122 clwlknon2num 28150 numclwlk1lem1 28151 numclwlk1lem2 28152 friendshipgt3 28180 |
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