Theorem upgredg 15957

Description: For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.) (Proof shortened by AV, 26-Nov-2021.)

Hypotheses

Ref	Expression
upgredg.v	⊢ 𝑉 = (Vtx‘𝐺)
upgredg.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
upgredg	⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})

Distinct variable groups:   𝐶,𝑎,𝑏   𝐺,𝑎,𝑏   𝑉,𝑎,𝑏

Allowed substitution hints:   𝐸(𝑎,𝑏)

Proof of Theorem upgredg

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgredg.e	. . . . 5 ⊢ 𝐸 = (Edg‘𝐺)
2		edgvalg 15875	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	1, 2	eqtrid 2274	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
4	3	eleq2d 2299	. . 3 ⊢ (𝐺 ∈ UPGraph → (𝐶 ∈ 𝐸 ↔ 𝐶 ∈ ran (iEdg‘𝐺)))
5	4	biimpa 296	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → 𝐶 ∈ ran (iEdg‘𝐺))
6		upgredg.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
7		eqid 2229	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	upgrfen 15912	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
9	8	ffnd 5474	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
10	9	adantr 276	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
11		fnfun 5418	. . . . 5 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → Fun (iEdg‘𝐺))
12	10, 11	syl 14	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) → Fun (iEdg‘𝐺))
13		simpr 110	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) → 𝐶 ∈ ran (iEdg‘𝐺))
14		elrnrexdm 5776	. . . 4 ⊢ (Fun (iEdg‘𝐺) → (𝐶 ∈ ran (iEdg‘𝐺) → ∃𝑧 ∈ dom (iEdg‘𝐺)𝐶 = ((iEdg‘𝐺)‘𝑧)))
15	12, 13, 14	sylc 62	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) → ∃𝑧 ∈ dom (iEdg‘𝐺)𝐶 = ((iEdg‘𝐺)‘𝑧))
16		simpll 527	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → 𝐺 ∈ UPGraph)
17	16, 9	syl 14	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
18		simprl 529	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → 𝑧 ∈ dom (iEdg‘𝐺))
19	6, 7	upgrex 15918	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (iEdg‘𝐺) Fn dom (iEdg‘𝐺) ∧ 𝑧 ∈ dom (iEdg‘𝐺)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ((iEdg‘𝐺)‘𝑧) = {𝑎, 𝑏})
20	16, 17, 18, 19	syl3anc 1271	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ((iEdg‘𝐺)‘𝑧) = {𝑎, 𝑏})
21		simprr 531	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → 𝐶 = ((iEdg‘𝐺)‘𝑧))
22	21	eqeq1d 2238	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → (𝐶 = {𝑎, 𝑏} ↔ ((iEdg‘𝐺)‘𝑧) = {𝑎, 𝑏}))
23	22	2rexbidv 2555	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → (∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏} ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ((iEdg‘𝐺)‘𝑧) = {𝑎, 𝑏}))
24	20, 23	mpbird 167	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})
25	15, 24	rexlimddv 2653	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})
26	5, 25	syldan 282	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝐶 = {𝑎, 𝑏})

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  {crab 2512  𝒫 cpw 3649  {cpr 3667   class class class wbr 4083  dom cdm 4719  ran crn 4720  Fun wfun 5312   Fn wfn 5313  ‘cfv 5318  1_oc1o 6561  2_oc2o 6562   ≈ cen 6893  Vtxcvtx 15828  iEdgciedg 15829  Edgcedg 15873  UPGraphcupgr 15906

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 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-1o 6568  df-2o 6569  df-en 6896  df-sub 8330  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-dec 9590  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831  df-edg 15874  df-upgren 15908

This theorem is referenced by:  upgrpredgv  15959  upgredg2vtx  15961  upgredgpr  15962