Theorem upgr1eopALT 28904

Description: Alternate proof of upgr1eop 28902, using the general theorem gropeld 28820 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 28902). (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 2727	. . . . 5 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
2		simpllr 775	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐴 ∈ 𝑋)
3		simplrl 776	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵 ∈ 𝑉)
4		eleq2 2817	. . . . . . 7 ⊢ ((Vtx‘𝑔) = 𝑉 → (𝐵 ∈ (Vtx‘𝑔) ↔ 𝐵 ∈ 𝑉))
5	4	ad2antrl 727	. . . . . 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 257	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵 ∈ (Vtx‘𝑔))
7		simplrr 777	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶 ∈ 𝑉)
8		eleq2 2817	. . . . . . 7 ⊢ ((Vtx‘𝑔) = 𝑉 → (𝐶 ∈ (Vtx‘𝑔) ↔ 𝐶 ∈ 𝑉))
9	8	ad2antrl 727	. . . . . 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 257	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶 ∈ (Vtx‘𝑔))
11		simprr 772	. . . . 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 28900	. . . 4 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝑔 ∈ UPGraph)
13	12	ex 412	. . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph))
14	13	alrimiv 1923	. 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph))
15		simpll 766	. 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑉 ∈ 𝑊)
16		snex 5427	. . 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 28820	1 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3469  {csn 4624  {cpr 4626  ⟨cop 4630  ‘cfv 6542  Vtxcvtx 28783  iEdgciedg 28784  UPGraphcupgr 28867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-oadd 8482  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-dju 9910  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-n0 12489  df-xnn0 12561  df-z 12575  df-uz 12839  df-fz 13503  df-hash 14308  df-vtx 28785  df-iedg 28786  df-upgr 28869

This theorem is referenced by: (None)