Theorem upgrex 16515

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 16513	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ((𝐸‘𝐹) ≈ 1o ∨ (𝐸‘𝐹) ≈ 2o))
4		en1 7086	. . . . . . 7 ⊢ ((𝐸‘𝐹) ≈ 1o ↔ ∃𝑧(𝐸‘𝐹) = {𝑧})
5		dfsn2 3723	. . . . . . . . 9 ⊢ {𝑧} = {𝑧, 𝑧}
6	5	eqeq2i 2249	. . . . . . . 8 ⊢ ((𝐸‘𝐹) = {𝑧} ↔ (𝐸‘𝐹) = {𝑧, 𝑧})
7	6	exbii 1658	. . . . . . 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 3789	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → {𝑧, 𝑦} = {𝑧, 𝑧})
10	9	eqeq2d 2250	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐸‘𝐹) = {𝑧, 𝑦} ↔ (𝐸‘𝐹) = {𝑧, 𝑧}))
11	10	spcegv 2913	. . . . . . . . 9 ⊢ (𝑧 ∈ V → ((𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦}))
12	11	elv 2825	. . . . . . . 8 ⊢ ((𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦})
13		preq1 3788	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥, 𝑦} = {𝑧, 𝑦})
14	13	eqeq2d 2250	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐸‘𝐹) = {𝑥, 𝑦} ↔ (𝐸‘𝐹) = {𝑧, 𝑦}))
15	14	exbidv 1878	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦} ↔ ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦}))
16	15	spcegv 2913	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦} → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦}))
17	16	elv 2825	. . . . . . . 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 1651	. . . . . 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 7112	. . . . 5 ⊢ ((𝐸‘𝐹) ≈ 2o → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
22	20, 21	jaoi 728	. . . 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 1028	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐺 ∈ UPGraph)
25		simp3 1030	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ 𝐴)
26		fndm 5480	. . . . . . . . . . 11 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
27	26	3ad2ant2 1050	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → dom 𝐸 = 𝐴)
28	25, 27	eleqtrrd 2318	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ dom 𝐸)
29	1, 2	upgrss 16511	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸‘𝐹) ⊆ 𝑉)
30	24, 28, 29	syl2anc 415	. . . . . . . 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 2824	. . . . . . . . 9 ⊢ 𝑥 ∈ V
33	32	prid1 3817	. . . . . . . 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 2328	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑥 ∈ (𝐸‘𝐹))
36	31, 35	sseldd 3249	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑥 ∈ 𝑉)
37		vex 2824	. . . . . . . . 9 ⊢ 𝑦 ∈ V
38	37	prid2 3818	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑥, 𝑦}
39	38, 34	eleqtrrid 2328	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑦 ∈ (𝐸‘𝐹))
40	31, 39	sseldd 3249	. . . . . 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 1935	. . 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 2570	. 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})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  Vcvv 2821   ⊆ wss 3220  {csn 3709  {cpr 3710   class class class wbr 4130  dom cdm 4774   Fn wfn 5372  ‘cfv 5377  1_oc1o 6680  2_oc2o 6681   ≈ cen 7020  Vtxcvtx 16424  iEdgciedg 16425  UPGraphcupgr 16503

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-en 7023  df-sub 8501  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-dec 9783  df-ndx 13407  df-slot 13408  df-base 13410  df-edgf 16417  df-vtx 16426  df-iedg 16427  df-upgren 16505

This theorem is used by:  upgredg  16556