Theorem constr3trllem1 26178

Description: Lemma for constr3trl 26187. (Contributed by Alexander van der Vekens, 10-Nov-2017.)

Hypotheses

Ref	Expression
constr3cycl.f	⊢ 𝐹 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}
constr3cycl.p	⊢ 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})

Assertion

Ref	Expression
constr3trllem1	⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹 ∈ Word dom 𝐸)

Proof of Theorem constr3trllem1

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		c0ex 9913	. . . . 5 ⊢ 0 ∈ V
2		1ex 9914	. . . . 5 ⊢ 1 ∈ V
3		2ex 10969	. . . . 5 ⊢ 2 ∈ V
4		fvex 6113	. . . . 5 ⊢ (◡𝐸‘{𝐴, 𝐵}) ∈ V
5		fvex 6113	. . . . 5 ⊢ (◡𝐸‘{𝐵, 𝐶}) ∈ V
6		fvex 6113	. . . . 5 ⊢ (◡𝐸‘{𝐶, 𝐴}) ∈ V
7		0ne1 10965	. . . . 5 ⊢ 0 ≠ 1
8		0ne2 11116	. . . . 5 ⊢ 0 ≠ 2
9		1ne2 11117	. . . . 5 ⊢ 1 ≠ 2
10	1, 2, 3, 4, 5, 6, 7, 8, 9	ftp 6329	. . . 4 ⊢ {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}:{0, 1, 2}⟶{(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})}
11		constr3cycl.f	. . . . . 6 ⊢ 𝐹 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}
12	11	a1i 11	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩})
13		fzo0to3tp 12421	. . . . . 6 ⊢ (0..^3) = {0, 1, 2}
14	13	a1i 11	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (0..^3) = {0, 1, 2})
15	12, 14	feq12d 5946	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹:(0..^3)⟶{(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})} ↔ {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}:{0, 1, 2}⟶{(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})}))
16	10, 15	mpbiri 247	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹:(0..^3)⟶{(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})})
17		usgraf 25875	. . . . . 6 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
18		f1f1orn 6061	. . . . . 6 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
19		f1ocnvdm 6440	. . . . . . . . 9 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (◡𝐸‘{𝐴, 𝐵}) ∈ dom 𝐸)
20	19	3ad2antr1 1219	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (◡𝐸‘{𝐴, 𝐵}) ∈ dom 𝐸)
21		f1ocnvdm 6440	. . . . . . . . 9 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (◡𝐸‘{𝐵, 𝐶}) ∈ dom 𝐸)
22	21	3ad2antr2 1220	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (◡𝐸‘{𝐵, 𝐶}) ∈ dom 𝐸)
23		f1ocnvdm 6440	. . . . . . . . 9 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (◡𝐸‘{𝐶, 𝐴}) ∈ dom 𝐸)
24	23	3ad2antr3 1221	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (◡𝐸‘{𝐶, 𝐴}) ∈ dom 𝐸)
25	20, 22, 24	3jca 1235	. . . . . . 7 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((◡𝐸‘{𝐴, 𝐵}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐵, 𝐶}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐶, 𝐴}) ∈ dom 𝐸))
26	25	ex 449	. . . . . 6 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ((◡𝐸‘{𝐴, 𝐵}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐵, 𝐶}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐶, 𝐴}) ∈ dom 𝐸)))
27	17, 18, 26	3syl 18	. . . . 5 ⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → ((◡𝐸‘{𝐴, 𝐵}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐵, 𝐶}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐶, 𝐴}) ∈ dom 𝐸)))
28	27	imp 444	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((◡𝐸‘{𝐴, 𝐵}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐵, 𝐶}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐶, 𝐴}) ∈ dom 𝐸))
29	4, 5, 6	tpss 4308	. . . 4 ⊢ (((◡𝐸‘{𝐴, 𝐵}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐵, 𝐶}) ∈ dom 𝐸 ∧ (◡𝐸‘{𝐶, 𝐴}) ∈ dom 𝐸) ↔ {(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})} ⊆ dom 𝐸)
30	28, 29	sylib 207	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → {(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})} ⊆ dom 𝐸)
31	16, 30	jca 553	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹:(0..^3)⟶{(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})} ∧ {(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})} ⊆ dom 𝐸))
32		fss 5969	. 2 ⊢ ((𝐹:(0..^3)⟶{(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})} ∧ {(◡𝐸‘{𝐴, 𝐵}), (◡𝐸‘{𝐵, 𝐶}), (◡𝐸‘{𝐶, 𝐴})} ⊆ dom 𝐸) → 𝐹:(0..^3)⟶dom 𝐸)
33		iswrdi 13164	. 2 ⊢ (𝐹:(0..^3)⟶dom 𝐸 → 𝐹 ∈ Word dom 𝐸)
34	31, 32, 33	3syl 18	1 ⊢ ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹 ∈ Word dom 𝐸)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ^◡ccnv 5037  dom cdm 5038  ran crn 5039  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  3c3 10948  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-word 13154  df-usgra 25862

This theorem is referenced by:  constr3trllem2  26179  constr3trl  26187