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| 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 2736 | . . 3 ⊢ (Vtx‘〈𝑉, 𝐸〉) = (Vtx‘〈𝑉, 𝐸〉) | |
| 2 | 1 | isfusgr 29336 | . 2 ⊢ (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ (Vtx‘〈𝑉, 𝐸〉) ∈ Fin)) | 
| 3 | opvtxfv 29022 | . . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘〈𝑉, 𝐸〉) = 𝑉) | |
| 4 | 3 | eleq1d 2825 | . . 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 | bitrid 283 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (〈𝑉, 𝐸〉 ∈ FinUSGraph ↔ (〈𝑉, 𝐸〉 ∈ USGraph ∧ 𝑉 ∈ Fin))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2107 〈cop 4631 ‘cfv 6560 Fincfn 8986 Vtxcvtx 29014 USGraphcusgr 29167 FinUSGraphcfusgr 29334 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fv 6568 df-1st 8015 df-vtx 29016 df-fusgr 29335 | 
| This theorem is referenced by: fusgrfis 29348 | 
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