Theorem upgrex 15947

Description: An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isupgr.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
upgrex	⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})

Distinct variable groups:   𝑥,𝐺   𝑥,𝑉   𝑥,𝐸   𝑥,𝐹   𝑥,𝐴,𝑦   𝑦,𝐸   𝑦,𝐹   𝑦,𝐺   𝑦,𝑉

Proof of Theorem upgrex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isupgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		isupgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	upgr1or2 15945	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ((𝐸‘𝐹) ≈ 1o ∨ (𝐸‘𝐹) ≈ 2o))
4		en1 6968	. . . . . . 7 ⊢ ((𝐸‘𝐹) ≈ 1o ↔ ∃𝑧(𝐸‘𝐹) = {𝑧})
5		dfsn2 3681	. . . . . . . . 9 ⊢ {𝑧} = {𝑧, 𝑧}
6	5	eqeq2i 2240	. . . . . . . 8 ⊢ ((𝐸‘𝐹) = {𝑧} ↔ (𝐸‘𝐹) = {𝑧, 𝑧})
7	6	exbii 1651	. . . . . . 7 ⊢ (∃𝑧(𝐸‘𝐹) = {𝑧} ↔ ∃𝑧(𝐸‘𝐹) = {𝑧, 𝑧})
8	4, 7	bitri 184	. . . . . 6 ⊢ ((𝐸‘𝐹) ≈ 1o ↔ ∃𝑧(𝐸‘𝐹) = {𝑧, 𝑧})
9		preq2 3747	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → {𝑧, 𝑦} = {𝑧, 𝑧})
10	9	eqeq2d 2241	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐸‘𝐹) = {𝑧, 𝑦} ↔ (𝐸‘𝐹) = {𝑧, 𝑧}))
11	10	spcegv 2892	. . . . . . . . 9 ⊢ (𝑧 ∈ V → ((𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦}))
12	11	elv 2804	. . . . . . . 8 ⊢ ((𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦})
13		preq1 3746	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥, 𝑦} = {𝑧, 𝑦})
14	13	eqeq2d 2241	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐸‘𝐹) = {𝑥, 𝑦} ↔ (𝐸‘𝐹) = {𝑧, 𝑦}))
15	14	exbidv 1871	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦} ↔ ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦}))
16	15	spcegv 2892	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦} → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦}))
17	16	elv 2804	. . . . . . . 8 ⊢ (∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦} → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
18	12, 17	syl 14	. . . . . . 7 ⊢ ((𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
19	18	exlimiv 1644	. . . . . 6 ⊢ (∃𝑧(𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
20	8, 19	sylbi 121	. . . . 5 ⊢ ((𝐸‘𝐹) ≈ 1o → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
21		en2 6993	. . . . 5 ⊢ ((𝐸‘𝐹) ≈ 2o → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
22	20, 21	jaoi 721	. . . 4 ⊢ (((𝐸‘𝐹) ≈ 1o ∨ (𝐸‘𝐹) ≈ 2o) → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
23	3, 22	syl 14	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
24		simp1 1021	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐺 ∈ UPGraph)
25		simp3 1023	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ 𝐴)
26		fndm 5426	. . . . . . . . . . 11 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
27	26	3ad2ant2 1043	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → dom 𝐸 = 𝐴)
28	25, 27	eleqtrrd 2309	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ dom 𝐸)
29	1, 2	upgrss 15943	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉)
30	24, 28, 29	syl2anc 411	. . . . . . . 8 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ⊆ 𝑉)
31	30	adantr 276	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → (𝐸‘𝐹) ⊆ 𝑉)
32		vex 2803	. . . . . . . . 9 ⊢ 𝑥 ∈ V
33	32	prid1 3775	. . . . . . . 8 ⊢ 𝑥 ∈ {𝑥, 𝑦}
34		simpr 110	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → (𝐸‘𝐹) = {𝑥, 𝑦})
35	33, 34	eleqtrrid 2319	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑥 ∈ (𝐸‘𝐹))
36	31, 35	sseldd 3226	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑥 ∈ 𝑉)
37		vex 2803	. . . . . . . . 9 ⊢ 𝑦 ∈ V
38	37	prid2 3776	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑥, 𝑦}
39	38, 34	eleqtrrid 2319	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑦 ∈ (𝐸‘𝐹))
40	31, 39	sseldd 3226	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑦 ∈ 𝑉)
41	36, 40, 34	jca31 309	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}))
42	41	ex 115	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ((𝐸‘𝐹) = {𝑥, 𝑦} → ((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝐸‘𝐹) = {𝑥, 𝑦})))
43	42	2eximdv 1928	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦} → ∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝐸‘𝐹) = {𝑥, 𝑦})))
44	23, 43	mpd 13	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}))
45		r2ex 2550	. 2 ⊢ (∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦} ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}))
46	44, 45	sylibr 134	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ 𝑉 (𝐸‘𝐹) = {𝑥, 𝑦})

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 713   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509  Vcvv 2800   ⊆ wss 3198  {csn 3667  {cpr 3668   class class class wbr 4086  dom cdm 4723   Fn wfn 5319  ‘cfv 5324  1_oc1o 6570  2_oc2o 6571   ≈ cen 6902  Vtxcvtx 15856  iEdgciedg 15857  UPGraphcupgr 15935

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 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136

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 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-1o 6577  df-2o 6578  df-en 6905  df-sub 8345  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-dec 9605  df-ndx 13078  df-slot 13079  df-base 13081  df-edgf 15849  df-vtx 15858  df-iedg 15859  df-upgren 15937

This theorem is referenced by:  upgredg  15988