Theorem upgrop 15889

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 2229	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2229	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrfen 15882	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)})
4		vtxex 15804	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Vtx‘𝐺) ∈ V)
5		iedgex 15805	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) ∈ V)
6		opexg 4313	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V)
7	4, 5, 6	syl2anc 411	. . . 4 ⊢ (𝐺 ∈ UPGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ V)
8		eqid 2229	. . . . 5 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
9		eqid 2229	. . . . 5 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
10	8, 9	isupgren 15880	. . . 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 15811	. . . . 5 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
13	4, 5, 12	syl2anc 411	. . . 4 ⊢ (𝐺 ∈ UPGraph → (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘𝐺))
14	13	dmeqd 4922	. . . 4 ⊢ (𝐺 ∈ UPGraph → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
15		opvtxfv 15808	. . . . . . 7 ⊢ (((Vtx‘𝐺) ∈ V ∧ (iEdg‘𝐺) ∈ V) → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
16	4, 5, 15	syl2anc 411	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘𝐺))
17	16	pweqd 3654	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = 𝒫 (Vtx‘𝐺))
18	17	rabeqdv 2793	. . . 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 5460	. . 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)

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200  {crab 2512  Vcvv 2799  𝒫 cpw 3649  ⟨cop 3669   class class class wbr 4082  dom cdm 4716  ⟶wf 5310  ‘cfv 5314  1_oc1o 6545  2_oc2o 6546   ≈ cen 6875  Vtxcvtx 15798  iEdgciedg 15799  UPGraphcupgr 15876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fo 5320  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-sub 8307  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-dec 9567  df-ndx 13021  df-slot 13022  df-base 13024  df-edgf 15791  df-vtx 15800  df-iedg 15801  df-upgren 15878

This theorem is referenced by: (None)