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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.
StepHypRef 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
101, 2, 3, 4, 5, 6, 7, 8, 9ftp 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})⟩}
1211a1i 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}
1413a1i 11 . . . . 5 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (0..^3) = {0, 1, 2})
1512, 14feq12d 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})}))
1610, 15mpbiri 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 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})
18 f1f1orn 6061 . . . . . 6 (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2} → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
19 f1ocnvdm 6440 . . . . . . . . 9 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐸‘{𝐴, 𝐵}) ∈ dom 𝐸)
20193ad2antr1 1219 . . . . . . . 8 ((𝐸:dom 𝐸1-1-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 𝐸1-1-onto→ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐸‘{𝐵, 𝐶}) ∈ dom 𝐸)
22213ad2antr2 1220 . . . . . . . 8 ((𝐸:dom 𝐸1-1-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 𝐸1-1-onto→ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸) → (𝐸‘{𝐶, 𝐴}) ∈ dom 𝐸)
24233ad2antr3 1221 . . . . . . . 8 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐸‘{𝐶, 𝐴}) ∈ dom 𝐸)
2520, 22, 243jca 1235 . . . . . . 7 ((𝐸:dom 𝐸1-1-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))
2625ex 449 . . . . . 6 (𝐸:dom 𝐸1-1-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)))
2717, 18, 263syl 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)))
2827imp 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))
294, 5, 6tpss 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)
3028, 29sylib 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)
3116, 30jca 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 𝐸𝐹 ∈ Word dom 𝐸)
3431, 32, 333syl 18 1 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → 𝐹 ∈ Word dom 𝐸)
 Colors of variables: wff setvar class 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
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