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Theorem upgr1eopALT 25941
Description: Alternate proof of upgr1eop 25939, using the general theorem gropeld 25859 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 25939). (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 2621 . . . . 5 (Vtx‘𝑔) = (Vtx‘𝑔)
2 simpllr 798 . . . . 5 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐴𝑋)
3 simplrl 799 . . . . . 6 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵𝑉)
4 eleq2 2687 . . . . . . 7 ((Vtx‘𝑔) = 𝑉 → (𝐵 ∈ (Vtx‘𝑔) ↔ 𝐵𝑉))
54ad2antrl 763 . . . . . 6 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → (𝐵 ∈ (Vtx‘𝑔) ↔ 𝐵𝑉))
63, 5mpbird 247 . . . . 5 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵 ∈ (Vtx‘𝑔))
7 simplrr 800 . . . . . 6 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶𝑉)
8 eleq2 2687 . . . . . . 7 ((Vtx‘𝑔) = 𝑉 → (𝐶 ∈ (Vtx‘𝑔) ↔ 𝐶𝑉))
98ad2antrl 763 . . . . . 6 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → (𝐶 ∈ (Vtx‘𝑔) ↔ 𝐶𝑉))
107, 9mpbird 247 . . . . 5 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶 ∈ (Vtx‘𝑔))
11 simprr 795 . . . . 5 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})
121, 2, 6, 10, 11upgr1e 25937 . . . 4 ((((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝑔 ∈ UPGraph )
1312ex 450 . . 3 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph ))
1413alrimiv 1852 . 2 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph ))
15 simpll 789 . 2 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → 𝑉𝑊)
16 snex 4879 . . 3 {⟨𝐴, {𝐵, 𝐶}⟩} ∈ V
1716a1i 11 . 2 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → {⟨𝐴, {𝐵, 𝐶}⟩} ∈ V)
1814, 15, 17gropeld 25859 1 (((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  {csn 4155  {cpr 4157  cop 4161  cfv 5857  Vtxcvtx 25808  iEdgciedg 25809   UPGraph cupgr 25905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074  df-vtx 25810  df-iedg 25811  df-upgr 25907
This theorem is referenced by: (None)
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