Theorem upgrex 15749

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 15747	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ((𝐸‘𝐹) ≈ 1o ∨ (𝐸‘𝐹) ≈ 2o))
4		en1 6901	. . . . . . 7 ⊢ ((𝐸‘𝐹) ≈ 1o ↔ ∃𝑧(𝐸‘𝐹) = {𝑧})
5		dfsn2 3649	. . . . . . . . 9 ⊢ {𝑧} = {𝑧, 𝑧}
6	5	eqeq2i 2217	. . . . . . . 8 ⊢ ((𝐸‘𝐹) = {𝑧} ↔ (𝐸‘𝐹) = {𝑧, 𝑧})
7	6	exbii 1629	. . . . . . 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 3713	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → {𝑧, 𝑦} = {𝑧, 𝑧})
10	9	eqeq2d 2218	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐸‘𝐹) = {𝑧, 𝑦} ↔ (𝐸‘𝐹) = {𝑧, 𝑧}))
11	10	spcegv 2863	. . . . . . . . 9 ⊢ (𝑧 ∈ V → ((𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦}))
12	11	elv 2777	. . . . . . . 8 ⊢ ((𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦})
13		preq1 3712	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥, 𝑦} = {𝑧, 𝑦})
14	13	eqeq2d 2218	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐸‘𝐹) = {𝑥, 𝑦} ↔ (𝐸‘𝐹) = {𝑧, 𝑦}))
15	14	exbidv 1849	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦} ↔ ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦}))
16	15	spcegv 2863	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦} → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦}))
17	16	elv 2777	. . . . . . . 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 1622	. . . . . 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 6923	. . . . 5 ⊢ ((𝐸‘𝐹) ≈ 2o → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
22	20, 21	jaoi 718	. . . 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 1000	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐺 ∈ UPGraph)
25		simp3 1002	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ 𝐴)
26		fndm 5379	. . . . . . . . . . 11 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
27	26	3ad2ant2 1022	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → dom 𝐸 = 𝐴)
28	25, 27	eleqtrrd 2286	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ dom 𝐸)
29	1, 2	upgrss 15745	. . . . . . . . 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 2776	. . . . . . . . 9 ⊢ 𝑥 ∈ V
33	32	prid1 3741	. . . . . . . 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 2296	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑥 ∈ (𝐸‘𝐹))
36	31, 35	sseldd 3196	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑥 ∈ 𝑉)
37		vex 2776	. . . . . . . . 9 ⊢ 𝑦 ∈ V
38	37	prid2 3742	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑥, 𝑦}
39	38, 34	eleqtrrid 2296	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑦 ∈ (𝐸‘𝐹))
40	31, 39	sseldd 3196	. . . . . 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 1906	. . 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 2527	. 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 710   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∃wrex 2486  Vcvv 2773   ⊆ wss 3168  {csn 3635  {cpr 3636   class class class wbr 4048  dom cdm 4680   Fn wfn 5272  ‘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-nul 4175  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-tr 4148  df-id 4345  df-iord 4418  df-on 4420  df-suc 4423  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-1o 6512  df-2o 6513  df-en 6838  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)