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Mirrors > Home > MPE Home > Th. List > opfusgr | Structured version Visualization version GIF version |
Description: A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.) |
Ref | Expression |
---|---|
opfusgr | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . 3 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
2 | 1 | isfusgr 27102 | . 2 ⊢ (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ (Vtx‘〈𝑉, 𝐸〉) ∈ Fin)) |
3 | opvtxfv 26791 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
4 | 3 | eleq1d 2899 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((Vtx‘〈𝑉, 𝐸〉) ∈ Fin ↔ 𝑉 ∈ Fin)) |
5 | 4 | anbi2d 630 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ((〈𝑉, 𝐸〉 ∈ USGraph ∧ (Vtx‘〈𝑉, 𝐸〉) ∈ Fin) ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) |
6 | 2, 5 | syl5bb 285 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 〈cop 4575 ‘cfv 6357 Fincfn 8511 Vtxcvtx 26783 USGraphcusgr 26936 FinUSGraphcfusgr 27100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-1st 7691 df-vtx 26785 df-fusgr 27101 |
This theorem is referenced by: fusgrfis 27114 |
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