Theorem upgredg 15818

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 15741	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	1, 2	eqtrid 2251	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
4	3	eleq2d 2276	. . 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 2206	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	upgrfen 15778	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
9	8	ffnd 5441	. . . . . 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 5385	. . . . 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 5737	. . . 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 15784	. . . . 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 1250	. . . 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 2215	. . . . 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 2532	. . . 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 2629	. 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 710   = wceq 1373   ∈ wcel 2177  ∃wrex 2486  {crab 2489  𝒫 cpw 3621  {cpr 3639   class class class wbr 4054  dom cdm 4688  ran crn 4689  Fun wfun 5279   Fn wfn 5280  ‘cfv 5285  1_oc1o 6513  2_oc2o 6514   ≈ cen 6843  Vtxcvtx 15696  iEdgciedg 15697  Edgcedg 15739  UPGraphcupgr 15772

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 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-cnre 8066

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 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-iord 4426  df-on 4428  df-suc 4431  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-1o 6520  df-2o 6521  df-en 6846  df-sub 8275  df-inn 9067  df-2 9125  df-3 9126  df-4 9127  df-5 9128  df-6 9129  df-7 9130  df-8 9131  df-9 9132  df-n0 9326  df-dec 9535  df-ndx 12920  df-slot 12921  df-base 12923  df-edgf 15689  df-vtx 15698  df-iedg 15699  df-edg 15740  df-upgren 15774

This theorem is referenced by:  upgrpredgv  15820  upgredg2vtx  15822  upgredgpr  15823