Theorem upgr1eopALT 29204

Description: Alternate proof of upgr1eop 29202, using the general theorem gropeld 29120 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of upgr1eop 29202). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
upgr1eopALT	⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph)

Proof of Theorem upgr1eopALT

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2739	. . . . 5 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
2		simpllr 781	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐴 ∈ 𝑋)
3		simplrl 782	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵 ∈ 𝑉)
4		eleq2 2828	. . . . . . 7 ⊢ ((Vtx‘𝑔) = 𝑉 → (𝐵 ∈ (Vtx‘𝑔) ↔ 𝐵 ∈ 𝑉))
5	4	ad2antrl 734	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → (𝐵 ∈ (Vtx‘𝑔) ↔ 𝐵 ∈ 𝑉))
6	3, 5	mpbird 258	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵 ∈ (Vtx‘𝑔))
7		simplrr 783	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶 ∈ 𝑉)
8		eleq2 2828	. . . . . . 7 ⊢ ((Vtx‘𝑔) = 𝑉 → (𝐶 ∈ (Vtx‘𝑔) ↔ 𝐶 ∈ 𝑉))
9	8	ad2antrl 734	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → (𝐶 ∈ (Vtx‘𝑔) ↔ 𝐶 ∈ 𝑉))
10	7, 9	mpbird 258	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶 ∈ (Vtx‘𝑔))
11		simprr 778	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})
12	1, 2, 6, 10, 11	upgr1e 29200	. . . 4 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝑔 ∈ UPGraph)
13	12	ex 413	. . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph))
14	13	alrimiv 1934	. 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph))
15		simpll 772	. 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑉 ∈ 𝑊)
16		snex 5368	. . 3 ⊢ {⟨𝐴, {𝐵, 𝐶}⟩} ∈ V
17	16	a1i 11	. 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → {⟨𝐴, {𝐵, 𝐶}⟩} ∈ V)
18	14, 15, 17	gropeld 29120	1 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  {csn 4555  {cpr 4557  ⟨cop 4561  ‘cfv 6485  Vtxcvtx 29083  iEdgciedg 29084  UPGraphcupgr 29167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-xnn0 12502  df-z 12516  df-uz 12780  df-fz 13453  df-hash 14284  df-vtx 29085  df-iedg 29086  df-upgr 29169

This theorem is referenced by: (None)