Theorem upgrex 16090

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 16088	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → ((𝐸‘𝐹) ≈ 1o ∨ (𝐸‘𝐹) ≈ 2o))
4		en1 7038	. . . . . . 7 ⊢ ((𝐸‘𝐹) ≈ 1o ↔ ∃𝑧(𝐸‘𝐹) = {𝑧})
5		dfsn2 3702	. . . . . . . . 9 ⊢ {𝑧} = {𝑧, 𝑧}
6	5	eqeq2i 2243	. . . . . . . 8 ⊢ ((𝐸‘𝐹) = {𝑧} ↔ (𝐸‘𝐹) = {𝑧, 𝑧})
7	6	exbii 1654	. . . . . . 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 3768	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → {𝑧, 𝑦} = {𝑧, 𝑧})
10	9	eqeq2d 2244	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐸‘𝐹) = {𝑧, 𝑦} ↔ (𝐸‘𝐹) = {𝑧, 𝑧}))
11	10	spcegv 2904	. . . . . . . . 9 ⊢ (𝑧 ∈ V → ((𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦}))
12	11	elv 2816	. . . . . . . 8 ⊢ ((𝐸‘𝐹) = {𝑧, 𝑧} → ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦})
13		preq1 3767	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → {𝑥, 𝑦} = {𝑧, 𝑦})
14	13	eqeq2d 2244	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐸‘𝐹) = {𝑥, 𝑦} ↔ (𝐸‘𝐹) = {𝑧, 𝑦}))
15	14	exbidv 1874	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦} ↔ ∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦}))
16	15	spcegv 2904	. . . . . . . . 9 ⊢ (𝑧 ∈ V → (∃𝑦(𝐸‘𝐹) = {𝑧, 𝑦} → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦}))
17	16	elv 2816	. . . . . . . 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 1647	. . . . . 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 7064	. . . . 5 ⊢ ((𝐸‘𝐹) ≈ 2o → ∃𝑥∃𝑦(𝐸‘𝐹) = {𝑥, 𝑦})
22	20, 21	jaoi 724	. . . 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 1024	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐺 ∈ UPGraph)
25		simp3 1026	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ 𝐴)
26		fndm 5454	. . . . . . . . . . 11 ⊢ (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
27	26	3ad2ant2 1046	. . . . . . . . . 10 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → dom 𝐸 = 𝐴)
28	25, 27	eleqtrrd 2312	. . . . . . . . 9 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → 𝐹 ∈ dom 𝐸)
29	1, 2	upgrss 16086	. . . . . . . . 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 2815	. . . . . . . . 9 ⊢ 𝑥 ∈ V
33	32	prid1 3796	. . . . . . . 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 2322	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑥 ∈ (𝐸‘𝐹))
36	31, 35	sseldd 3238	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑥 ∈ 𝑉)
37		vex 2815	. . . . . . . . 9 ⊢ 𝑦 ∈ V
38	37	prid2 3797	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑥, 𝑦}
39	38, 34	eleqtrrid 2322	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) ∧ (𝐸‘𝐹) = {𝑥, 𝑦}) → 𝑦 ∈ (𝐸‘𝐹))
40	31, 39	sseldd 3238	. . . . . 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 1931	. . 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 2562	. 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 716   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∃wrex 2521  Vcvv 2812   ⊆ wss 3210  {csn 3688  {cpr 3689   class class class wbr 4108  dom cdm 4748   Fn wfn 5346  ‘cfv 5351  1_oc1o 6639  2_oc2o 6640   ≈ cen 6972  Vtxcvtx 15999  iEdgciedg 16000  UPGraphcupgr 16078

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-1o 6646  df-2o 6647  df-en 6975  df-sub 8445  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-5 9298  df-6 9299  df-7 9300  df-8 9301  df-9 9302  df-n0 9496  df-dec 9709  df-ndx 13207  df-slot 13208  df-base 13210  df-edgf 15992  df-vtx 16001  df-iedg 16002  df-upgren 16080

This theorem is referenced by:  upgredg  16131