Theorem upgrop 16516

Description: A pseudograph represented by an ordered pair. (Contributed by AV, 12-Dec-2021.)

Assertion

Ref	Expression
upgrop	⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)

Proof of Theorem upgrop

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2238	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrfen 16509	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)})
4		vtxex 16430	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Vtx‘𝐺) ∈ V)
5		iedgex 16431	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) ∈ V)
6		opexg 4368	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V)
7	4, 5, 6	syl2anc 415	. . . 4 ⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V)
8		eqid 2238	. . . . 5 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
9		eqid 2238	. . . . 5 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
10	8, 9	isupgren 16507	. . . 4 ⊢ (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V → (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ↔ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩):dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)⟶{𝑝 ∈ 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)}))
11	7, 10	syl 14	. . 3 ⊢ (𝐺 ∈ UPGraph → (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ↔ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩):dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)⟶{𝑝 ∈ 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)}))
12		opiedgfv 16437	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
13	4, 5, 12	syl2anc 415	. . . 4 ⊢ (𝐺 ∈ UPGraph → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
14	13	dmeqd 4983	. . . 4 ⊢ (𝐺 ∈ UPGraph → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
15		opvtxfv 16434	. . . . . . 7 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
16	4, 5, 15	syl2anc 415	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
17	16	pweqd 3693	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = 𝒫 (Vtx‘𝐺))
18	17	rabeqdv 2815	. . . 4 ⊢ (𝐺 ∈ UPGraph → {𝑝 ∈ 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)} = {𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)})
19	13, 14, 18	feq123d 5524	. . 3 ⊢ (𝐺 ∈ UPGraph → ((iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩):dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)⟶{𝑝 ∈ 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)} ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)}))
20	11, 19	bitrd 188	. 2 ⊢ (𝐺 ∈ UPGraph → (⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)}))
21	3, 20	mpbird 167	1 ⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821  𝒫 cpw 3688  ⟨cop 3712   class class class wbr 4130  dom cdm 4774  ⟶wf 5373  ‘cfv 5377  1_oc1o 6680  2_oc2o 6681   ≈ cen 7020  Vtxcvtx 16424  iEdgciedg 16425  UPGraphcupgr 16503

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-sub 8501  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-dec 9783  df-ndx 13407  df-slot 13408  df-base 13410  df-edgf 16417  df-vtx 16426  df-iedg 16427  df-upgren 16505

This theorem is used by: (None)