Theorem upgr0eopALT 29185

Description: Alternate proof of upgr0eop 29183, using the general theorem gropeld 29102 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 29183). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
upgr0eopALT	⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Proof of Theorem upgr0eopALT

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3433	. . . . . 6 ⊢ 𝑔 ∈ V
2	1	a1i 11	. . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ V)
3		simpr 484	. . . . 5 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → (iEdg‘𝑔) = ∅)
4	2, 3	upgr0e 29180	. . . 4 ⊢ (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
5	4	ax-gen 1797	. . 3 ⊢ ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph)
6	5	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = ∅) → 𝑔 ∈ UPGraph))
7		id 22	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ 𝑊)
8		0ex 5242	. . 3 ⊢ ∅ ∈ V
9	8	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → ∅ ∈ V)
10	6, 7, 9	gropeld 29102	1 ⊢ (𝑉 ∈ 𝑊 → ⟨𝑉, ∅⟩ ∈ UPGraph)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  Vcvv 3429  ∅c0 4273  ⟨cop 4573  ‘cfv 6498  Vtxcvtx 29065  iEdgciedg 29066  UPGraphcupgr 29149

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-i2m1 11106  ax-1ne0 11107  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-1st 7942  df-2nd 7943  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-2 12244  df-vtx 29067  df-iedg 29068  df-upgr 29151  df-umgr 29152

This theorem is referenced by: (None)