Theorem upgrop 15750

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 2206	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2206	. . 3 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	upgrfen 15743	. 2 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑝 ∈ 𝒫 (Vtx‘𝐺) ∣ (𝑝 ≈ 1o ∨ 𝑝 ≈ 2o)})
4		vtxex 15667	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Vtx‘𝐺) ∈ V)
5		iedgex 15668	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) ∈ V)
6		opexg 4277	. . . . 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 2206	. . . . 5 ⊢ (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
9		eqid 2206	. . . . 5 ⊢ (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩)
10	8, 9	isupgren 15741	. . . 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 15674	. . . . 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 4886	. . . 4 ⊢ (𝐺 ∈ UPGraph → dom (iEdg‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = dom (iEdg‘𝐺))
15		opvtxfv 15671	. . . . . . 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 3623	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝒫 (Vtx‘⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩) = 𝒫 (Vtx‘𝐺))
18	17	rabeqdv 2767	. . . 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 5423	. . 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 710   = wceq 1373   ∈ wcel 2177  {crab 2489  Vcvv 2773  𝒫 cpw 3618  ⟨cop 3638   class class class wbr 4048  dom cdm 4680  ⟶wf 5273  ‘cfv 5277  1_oc1o 6505  2_oc2o 6506   ≈ cen 6835  Vtxcvtx 15661  iEdgciedg 15662  UPGraphcupgr 15737

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-cnre 8049

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-fo 5283  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-sub 8258  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-5 9111  df-6 9112  df-7 9113  df-8 9114  df-9 9115  df-n0 9309  df-dec 9518  df-ndx 12885  df-slot 12886  df-base 12888  df-edgf 15654  df-vtx 15663  df-iedg 15664  df-upgren 15739

This theorem is referenced by: (None)