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Theorem upgr1eopALT 26914
 Description: Alternate proof of upgr1eop 26912, using the general theorem gropeld 26830 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 26912). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
upgr1eopALT (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph)

Proof of Theorem upgr1eopALT
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqid 2801 . . . . 5 (Vtx‘𝑔) = (Vtx‘𝑔)
2 simpllr 775 . . . . 5 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐴𝑋)
3 simplrl 776 . . . . . 6 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵𝑉)
4 eleq2 2881 . . . . . . 7 ((Vtx‘𝑔) = 𝑉 → (𝐵 ∈ (Vtx‘𝑔) ↔ 𝐵𝑉))
54ad2antrl 727 . . . . . 6 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → (𝐵 ∈ (Vtx‘𝑔) ↔ 𝐵𝑉))
63, 5mpbird 260 . . . . 5 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵 ∈ (Vtx‘𝑔))
7 simplrr 777 . . . . . 6 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶𝑉)
8 eleq2 2881 . . . . . . 7 ((Vtx‘𝑔) = 𝑉 → (𝐶 ∈ (Vtx‘𝑔) ↔ 𝐶𝑉))
98ad2antrl 727 . . . . . 6 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → (𝐶 ∈ (Vtx‘𝑔) ↔ 𝐶𝑉))
107, 9mpbird 260 . . . . 5 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶 ∈ (Vtx‘𝑔))
11 simprr 772 . . . . 5 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})
121, 2, 6, 10, 11upgr1e 26910 . . . 4 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝑔 ∈ UPGraph)
1312ex 416 . . 3 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph))
1413alrimiv 1928 . 2 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph))
15 simpll 766 . 2 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → 𝑉𝑊)
16 snex 5300 . . 3 {⟨𝐴, {𝐵, 𝐶}⟩} ∈ V
1716a1i 11 . 2 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → {⟨𝐴, {𝐵, 𝐶}⟩} ∈ V)
1814, 15, 17gropeld 26830 1 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  Vcvv 3444  {csn 4528  {cpr 4530  ⟨cop 4534  ‘cfv 6328  Vtxcvtx 26793  iEdgciedg 26794  UPGraphcupgr 26877 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12890  df-hash 13691  df-vtx 26795  df-iedg 26796  df-upgr 26879 This theorem is referenced by: (None)
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