Theorem opfusgr 29272

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

Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = (Vtx‘⟨𝑉, 𝐸⟩)
2	1	isfusgr 29267	. 2 ⊢ (⟨𝑉, 𝐸⟩ ∈ FinUSGraph ↔ (⟨𝑉, 𝐸⟩ ∈ USGraph ∧ (Vtx‘⟨𝑉, 𝐸⟩) ∈ Fin))
3		opvtxfv 28953	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
4	3	eleq1d 2813	. . 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)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ⟨cop 4583  ‘cfv 6482  Fincfn 8872  Vtxcvtx 28945  USGraphcusgr 29098  FinUSGraphcfusgr 29265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fv 6490  df-1st 7924  df-vtx 28947  df-fusgr 29266

This theorem is referenced by:  fusgrfis  29279