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Theorem upgr0eopALT 26231
 Description: Alternate proof of upgr0eop 26229, using the general theorem gropeld 26145 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 upgr0eop 26229). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
upgr0eopALT (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Proof of Theorem upgr0eopALT
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 vex 3352 . . . . . 6 𝑔 ∈ V
21a1i 11 . . . . 5 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V)
3 simpr 471 . . . . 5 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅)
42, 3upgr0e 26226 . . . 4 (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
54ax-gen 1869 . . 3 𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
65a1i 11 . 2 (𝑉𝑊 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph))
7 id 22 . 2 (𝑉𝑊𝑉𝑊)
8 0ex 4921 . . 3 ∅ ∈ V
98a1i 11 . 2 (𝑉𝑊 → ∅ ∈ V)
106, 7, 9gropeld 26145 1 (𝑉𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382  ∀wal 1628   = wceq 1630   ∈ wcel 2144  Vcvv 3349  ∅c0 4061  ⟨cop 4320  ‘cfv 6031  Vtxcvtx 26094  iEdgciedg 26095  UPGraphcupgr 26195 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-i2m1 10205  ax-1ne0 10206  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-1st 7314  df-2nd 7315  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-2 11280  df-vtx 26096  df-iedg 26097  df-upgr 26197  df-umgr 26198 This theorem is referenced by: (None)
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