Theorem upgredg 15994

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 15909	. . . . 5 ⊢ (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3	1, 2	eqtrid 2276	. . . 4 ⊢ (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
4	3	eleq2d 2301	. . 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 2231	. . . . . . . 8 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	upgrfen 15947	. . . . . . 7 ⊢ (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ 𝒫 𝑉 ∣ (𝑥 ≈ 1o ∨ 𝑥 ≈ 2o)})
9	8	ffnd 5483	. . . . . 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 5427	. . . . 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 5786	. . . 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 531	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → 𝑧 ∈ dom (iEdg‘𝐺))
19	6, 7	upgrex 15953	. . . . 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 1273	. . . 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 533	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐶 ∈ ran (iEdg‘𝐺)) ∧ (𝑧 ∈ dom (iEdg‘𝐺) ∧ 𝐶 = ((iEdg‘𝐺)‘𝑧))) → 𝐶 = ((iEdg‘𝐺)‘𝑧))
22	21	eqeq1d 2240	. . . . 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 2557	. . . 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 2655	. 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 715   = wceq 1397   ∈ wcel 2202  ∃wrex 2511  {crab 2514  𝒫 cpw 3652  {cpr 3670   class class class wbr 4088  dom cdm 4725  ran crn 4726  Fun wfun 5320   Fn wfn 5321  ‘cfv 5326  1_oc1o 6574  2_oc2o 6575   ≈ cen 6906  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907  UPGraphcupgr 15941

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-1o 6581  df-2o 6582  df-en 6909  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-upgren 15943

This theorem is referenced by:  upgrpredgv  15996  upgredg2vtx  15998  upgredgpr  15999  usgr1vr  16098