Theorem upgredg 16551

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 16466	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	1, 2	eqtrid 2283	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
4	3	eleq2d 2308	. . 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 2238	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	upgrfen 16504	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
9	8	ffnd 5534	. . . . . 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 5478	. . . . 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 5847	. . . 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 531	. . . . 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 535	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → 𝑧 ∈ dom (iEdg‘𝐺))
19	6, 7	upgrex 16510	. . . . 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 1278	. . . 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 537	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → 𝐶 = ((iEdg‘𝐺)‘𝑧))
22	21	eqeq1d 2247	. . . . 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 2575	. . . 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 2673	. 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})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  {crab 2532  𝒫 cpw 3688  {cpr 3710   class class class wbr 4130  dom cdm 4774  ran crn 4775  Fun wfun 5371   Fn wfn 5372  ‘cfv 5377  1_oc1o 6680  2_oc2o 6681   ≈ cen 7020  Vtxcvtx 16419  iEdgciedg 16420  Edgcedg 16464  UPGraphcupgr 16498

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 16412  df-vtx 16421  df-iedg 16422  df-edg 16465  df-upgren 16500

This theorem is used by:  upgrpredgv  16553  upgredg2vtx  16555  upgredgpr  16556  usgr1vr  16655