Theorem upgredg2vtx 16303

Description: For a vertex incident to an edge there is another vertex incident to the edge in a pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 5-Dec-2020.)

Hypotheses

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

Assertion

Ref	Expression
upgredg2vtx	⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏})

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

Allowed substitution hint:   𝐸(𝑏)

Proof of Theorem upgredg2vtx

Dummy variables 𝑎 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgredg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		upgredg.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3	1, 2	upgredg 16299	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸) → ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝐶 = {𝑎, 𝑐})
4	3	3adant3 1048	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → ∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝐶 = {𝑎, 𝑐})
5		elpr2elpr 3896	. . . . . . 7 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉 ∧ 𝐴 ∈ {𝑎, 𝑐}) → ∃𝑏 ∈ 𝑉 {𝑎, 𝑐} = {𝐴, 𝑏})
6	5	3expia 1236	. . . . . 6 ⊢ ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝐴 ∈ {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 {𝑎, 𝑐} = {𝐴, 𝑏}))
7		eleq2 2302	. . . . . . 7 ⊢ (𝐶 = {𝑎, 𝑐} → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ {𝑎, 𝑐}))
8		eqeq1 2245	. . . . . . . 8 ⊢ (𝐶 = {𝑎, 𝑐} → (𝐶 = {𝐴, 𝑏} ↔ {𝑎, 𝑐} = {𝐴, 𝑏}))
9	8	rexbidv 2551	. . . . . . 7 ⊢ (𝐶 = {𝑎, 𝑐} → (∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏} ↔ ∃𝑏 ∈ 𝑉 {𝑎, 𝑐} = {𝐴, 𝑏}))
10	7, 9	imbi12d 234	. . . . . 6 ⊢ (𝐶 = {𝑎, 𝑐} → ((𝐴 ∈ 𝐶 → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏}) ↔ (𝐴 ∈ {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 {𝑎, 𝑐} = {𝐴, 𝑏})))
11	6, 10	imbitrrid 156	. . . . 5 ⊢ (𝐶 = {𝑎, 𝑐} → ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝐴 ∈ 𝐶 → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏})))
12	11	com13 80	. . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝐶 = {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏})))
13	12	3ad2ant3 1051	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → ((𝑎 ∈ 𝑉 ∧ 𝑐 ∈ 𝑉) → (𝐶 = {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏})))
14	13	rexlimdvv 2675	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → (∃𝑎 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝐶 = {𝑎, 𝑐} → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏}))
15	4, 14	mpd 13	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐶 ∈ 𝐸 ∧ 𝐴 ∈ 𝐶) → ∃𝑏 ∈ 𝑉 𝐶 = {𝐴, 𝑏})

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  {cpr 3706  ‘cfv 5372  Vtxcvtx 16167  Edgcedg 16212  UPGraphcupgr 16246

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-1o 6677  df-2o 6678  df-en 7013  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-upgren 16248

This theorem is referenced by:  usgredg2vtx  16372  uspgredg2vtxeu  16373