Theorem upgr1eopALT 29688

Description: Alternate proof of upgr1eop 29686, using the general theorem gropeld 29604 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 29686). (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 2761	. . . . 5 ⊢ (Vtx‘𝑔) = (Vtx‘𝑔)
2		simpllr 788	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐴 ∈ 𝑋)
3		simplrl 789	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵 ∈ 𝑉)
4		eleq2 2850	. . . . . . 7 ⊢ ((Vtx‘𝑔) = 𝑉 → (𝐵 ∈ (Vtx‘𝑔) ↔ 𝐵 ∈ 𝑉))
5	4	ad2antrl 741	. . . . . 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 260	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐵 ∈ (Vtx‘𝑔))
7		simplrr 790	. . . . . 6 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶 ∈ 𝑉)
8		eleq2 2850	. . . . . . 7 ⊢ ((Vtx‘𝑔) = 𝑉 → (𝐶 ∈ (Vtx‘𝑔) ↔ 𝐶 ∈ 𝑉))
9	8	ad2antrl 741	. . . . . 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 260	. . . . 5 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝐶 ∈ (Vtx‘𝑔))
11		simprr 785	. . . . 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 29684	. . . 4 ⊢ ((((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩})) → 𝑔 ∈ UPGraph)
13	12	ex 418	. . 3 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph))
14	13	alrimiv 1960	. 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = {⟨𝐴, {𝐵, 𝐶}⟩}) → 𝑔 ∈ UPGraph))
15		simpll 779	. 2 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑉 ∈ 𝑊)
16		snex 5397	. . 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 29604	1 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑋) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ⟨𝑉, {⟨𝐴, {𝐵, 𝐶}⟩}⟩ ∈ UPGraph)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  {csn 4584  {cpr 4586  ⟨cop 4590  ‘cfv 6537  Vtxcvtx 29567  iEdgciedg 29568  UPGraphcupgr 29651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oadd 8473  df-er 8710  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-n0 12600  df-xnn0 12673  df-z 12687  df-uz 12959  df-fz 13633  df-hash 14468  df-vtx 29569  df-iedg 29570  df-upgr 29653

This theorem is used by: (None)