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Theorem opfusgr 29355
Description: A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.)
Assertion
Ref Expression
opfusgr ((𝑉𝑋𝐸𝑌) → (⟨𝑉, 𝐸⟩ ∈ FinUSGraph ↔ (⟨𝑉, 𝐸⟩ ∈ USGraph ∧ 𝑉 ∈ Fin)))

Proof of Theorem opfusgr
StepHypRef Expression
1 eqid 2735 . . 3 (Vtx‘⟨𝑉, 𝐸⟩) = (Vtx‘⟨𝑉, 𝐸⟩)
21isfusgr 29350 . 2 (⟨𝑉, 𝐸⟩ ∈ FinUSGraph ↔ (⟨𝑉, 𝐸⟩ ∈ USGraph ∧ (Vtx‘⟨𝑉, 𝐸⟩) ∈ Fin))
3 opvtxfv 29036 . . . 4 ((𝑉𝑋𝐸𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
43eleq1d 2824 . . 3 ((𝑉𝑋𝐸𝑌) → ((Vtx‘⟨𝑉, 𝐸⟩) ∈ Fin ↔ 𝑉 ∈ Fin))
54anbi2d 630 . 2 ((𝑉𝑋𝐸𝑌) → ((⟨𝑉, 𝐸⟩ ∈ USGraph ∧ (Vtx‘⟨𝑉, 𝐸⟩) ∈ Fin) ↔ (⟨𝑉, 𝐸⟩ ∈ USGraph ∧ 𝑉 ∈ Fin)))
62, 5bitrid 283 1 ((𝑉𝑋𝐸𝑌) → (⟨𝑉, 𝐸⟩ ∈ FinUSGraph ↔ (⟨𝑉, 𝐸⟩ ∈ USGraph ∧ 𝑉 ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  cop 4637  cfv 6563  Fincfn 8984  Vtxcvtx 29028  USGraphcusgr 29181  FinUSGraphcfusgr 29348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-vtx 29030  df-fusgr 29349
This theorem is referenced by:  fusgrfis  29362
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