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Theorem opfusgr 28580
Description: A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.)
Assertion
Ref Expression
opfusgr ((𝑉 ∈ 𝑋 ∧ 𝐞 ∈ 𝑌) → (⟚𝑉, 𝐞⟩ ∈ FinUSGraph ↔ (⟚𝑉, 𝐞⟩ ∈ USGraph ∧ 𝑉 ∈ Fin)))

Proof of Theorem opfusgr
StepHypRef Expression
1 eqid 2733 . . 3 (Vtx‘⟚𝑉, 𝐞⟩) = (Vtx‘⟚𝑉, 𝐞⟩)
21isfusgr 28575 . 2 (⟚𝑉, 𝐞⟩ ∈ FinUSGraph ↔ (⟚𝑉, 𝐞⟩ ∈ USGraph ∧ (Vtx‘⟚𝑉, 𝐞⟩) ∈ Fin))
3 opvtxfv 28264 . . . 4 ((𝑉 ∈ 𝑋 ∧ 𝐞 ∈ 𝑌) → (Vtx‘⟚𝑉, 𝐞⟩) = 𝑉)
43eleq1d 2819 . . 3 ((𝑉 ∈ 𝑋 ∧ 𝐞 ∈ 𝑌) → ((Vtx‘⟚𝑉, 𝐞⟩) ∈ Fin ↔ 𝑉 ∈ Fin))
54anbi2d 630 . 2 ((𝑉 ∈ 𝑋 ∧ 𝐞 ∈ 𝑌) → ((⟚𝑉, 𝐞⟩ ∈ USGraph ∧ (Vtx‘⟚𝑉, 𝐞⟩) ∈ Fin) ↔ (⟚𝑉, 𝐞⟩ ∈ USGraph ∧ 𝑉 ∈ Fin)))
62, 5bitrid 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 205   ∧ wa 397   ∈ wcel 2107  âŸšcop 4635  â€˜cfv 6544  Fincfn 8939  Vtxcvtx 28256  USGraphcusgr 28409  FinUSGraphcfusgr 28573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975  df-vtx 28258  df-fusgr 28574
This theorem is referenced by:  fusgrfis  28587
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