Theorem upgrunop 15768

Description: The union of two pseudographs (with the same vertex set): If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are pseudographs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.)

Hypotheses

Ref	Expression
upgrun.g	⊢ (𝜑 → 𝐺 ∈ UPGraph)
upgrun.h	⊢ (𝜑 → 𝐻 ∈ UPGraph)
upgrun.e	⊢ 𝐸 = (iEdg‘𝐺)
upgrun.f	⊢ 𝐹 = (iEdg‘𝐻)
upgrun.vg	⊢ 𝑉 = (Vtx‘𝐺)
upgrun.vh	⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
upgrun.i	⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)

Assertion

Ref	Expression
upgrunop	⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)

Proof of Theorem upgrunop

Step	Hyp	Ref	Expression
1		upgrun.g	. 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph)
2		upgrun.h	. 2 ⊢ (𝜑 → 𝐻 ∈ UPGraph)
3		upgrun.e	. 2 ⊢ 𝐸 = (iEdg‘𝐺)
4		upgrun.f	. 2 ⊢ 𝐹 = (iEdg‘𝐻)
5		upgrun.vg	. 2 ⊢ 𝑉 = (Vtx‘𝐺)
6		upgrun.vh	. 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉)
7		upgrun.i	. 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅)
8		vtxex 15667	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Vtx‘𝐺) ∈ V)
9	1, 8	syl 14	. . . 4 ⊢ (𝜑 → (Vtx‘𝐺) ∈ V)
10	5, 9	eqeltrid 2293	. . 3 ⊢ (𝜑 → 𝑉 ∈ V)
11		iedgex 15668	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) ∈ V)
12	1, 11	syl 14	. . . . 5 ⊢ (𝜑 → (iEdg‘𝐺) ∈ V)
13	3, 12	eqeltrid 2293	. . . 4 ⊢ (𝜑 → 𝐸 ∈ V)
14		iedgex 15668	. . . . . 6 ⊢ (𝐻 ∈ UPGraph → (iEdg‘𝐻) ∈ V)
15	2, 14	syl 14	. . . . 5 ⊢ (𝜑 → (iEdg‘𝐻) ∈ V)
16	4, 15	eqeltrid 2293	. . . 4 ⊢ (𝜑 → 𝐹 ∈ V)
17		unexg 4495	. . . 4 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ V) → (𝐸 ∪ 𝐹) ∈ V)
18	13, 16, 17	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐸 ∪ 𝐹) ∈ V)
19		opexg 4277	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V)
20	10, 18, 19	syl2anc 411	. 2 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V)
21		opvtxfv 15671	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (Vtx‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = 𝑉)
22	10, 18, 21	syl2anc 411	. 2 ⊢ (𝜑 → (Vtx‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = 𝑉)
23		opiedgfv 15674	. . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (iEdg‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = (𝐸 ∪ 𝐹))
24	10, 18, 23	syl2anc 411	. 2 ⊢ (𝜑 → (iEdg‘⟨𝑉, (𝐸 ∪ 𝐹)⟩) = (𝐸 ∪ 𝐹))
25	1, 2, 3, 4, 5, 6, 7, 20, 22, 24	upgrun 15767	1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2177  Vcvv 2773   ∪ cun 3166   ∩ cin 3167  ∅c0 3462  ⟨cop 3638  dom cdm 4680  ‘cfv 5277  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-nul 3463  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)