Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 6103 | . . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
2 | isfusgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 1, 2 | syl6eqr 2662 | . . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
4 | 3 | eleq1d 2672 | . 2 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∈ Fin ↔ 𝑉 ∈ Fin)) |
5 | df-fusgr 40536 | . 2 ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin} | |
6 | 4, 5 | elrab2 3333 | 1 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 Fincfn 7841 Vtxcvtx 25673 USGraph cusgr 40379 FinUSGraph cfusgr 40535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-fusgr 40536 |
This theorem is referenced by: fusgrvtxfi 40538 isfusgrf1 40539 isfusgrcl 40540 fusgrusgr 40541 opfusgr 40542 fusgredgfi 40544 fusgrfis 40549 nbfusgrlevtxm1 40605 nbfusgrlevtxm2 40606 cusgrsizeindslem 40667 cusgrsizeinds 40668 sizusglecusglem2 40678 fusgrmaxsize 40680 finrusgrfusgr 40765 rusgrnumwwlks 41177 rusgrnumwwlk 41178 frrusgrord0 41503 av-friendshipgt3 41552 |
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