Theorem upgrop 15912

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 15905	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)})
4		vtxex 15827	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Vtx‘𝐺) ∈ V)
5		iedgex 15828	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) ∈ V)
6		opexg 4314	. . . . 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 15903	. . . 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 15834	. . . . 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 4925	. . . 4 ⊢ (𝐺 ∈ UPGraph → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
15		opvtxfv 15831	. . . . . . 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 5464	. . 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 4083  dom cdm 4719  ⟶wf 5314  ‘cfv 5318  1_oc1o 6561  2_oc2o 6562   ≈ cen 6893  Vtxcvtx 15821  iEdgciedg 15822  UPGraphcupgr 15899

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 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-cnre 8118

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 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-sub 8327  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-dec 9587  df-ndx 13043  df-slot 13044  df-base 13046  df-edgf 15814  df-vtx 15823  df-iedg 15824  df-upgren 15901

This theorem is referenced by: (None)