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 Description: Lemma 2 for usgra2adedgspth 26141. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
Assertion
Ref Expression
usgra2adedgspthlem2 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}) ∧ (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))

StepHypRef Expression
1 usgraf1o 25887 . . . 4 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)
2 f1of1 6049 . . . . 5 (𝐸:dom 𝐸1-1-onto→ran 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
3 f1ocnvfvrneq 6441 . . . . . . . . . 10 ((𝐸:dom 𝐸1-1→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶}))
43adantlr 747 . . . . . . . . 9 (((𝐸:dom 𝐸1-1→ran 𝐸𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶}))
5 usgraedgrnv 25906 . . . . . . . . . . . . . . . . . . 19 ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐵𝑉𝐶𝑉))
6 usgraedgrnv 25906 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐴𝑉𝐵𝑉))
7 pm3.2 462 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴𝑉𝐵𝑉) → ((𝐵𝑉𝐶𝑉) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))))
86, 7syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ((𝐵𝑉𝐶𝑉) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))))
98ex 449 . . . . . . . . . . . . . . . . . . . 20 (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → ((𝐵𝑉𝐶𝑉) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)))))
109com13 86 . . . . . . . . . . . . . . . . . . 19 ((𝐵𝑉𝐶𝑉) → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)))))
115, 10syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)))))
1211ex 449 . . . . . . . . . . . . . . . . 17 (𝑉 USGrph 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))))))
1312com13 86 . . . . . . . . . . . . . . . 16 ({𝐴, 𝐵} ∈ ran 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))))))
1413imp 444 . . . . . . . . . . . . . . 15 (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)))))
1514com13 86 . . . . . . . . . . . . . 14 (𝑉 USGrph 𝐸 → (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)))))
1615pm2.43i 50 . . . . . . . . . . . . 13 (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))))
1716adantl 481 . . . . . . . . . . . 12 ((𝐸:dom 𝐸1-1→ran 𝐸𝑉 USGrph 𝐸) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉))))
1817imp 444 . . . . . . . . . . 11 (((𝐸:dom 𝐸1-1→ran 𝐸𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)))
19 preq12bg 4326 . . . . . . . . . . 11 (((𝐴𝑉𝐵𝑉) ∧ (𝐵𝑉𝐶𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
2018, 19syl 17 . . . . . . . . . 10 (((𝐸:dom 𝐸1-1→ran 𝐸𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵))))
21 eqtr 2629 . . . . . . . . . . 11 ((𝐴 = 𝐵𝐵 = 𝐶) → 𝐴 = 𝐶)
22 simpl 472 . . . . . . . . . . 11 ((𝐴 = 𝐶𝐵 = 𝐵) → 𝐴 = 𝐶)
2321, 22jaoi 393 . . . . . . . . . 10 (((𝐴 = 𝐵𝐵 = 𝐶) ∨ (𝐴 = 𝐶𝐵 = 𝐵)) → 𝐴 = 𝐶)
2420, 23syl6bi 242 . . . . . . . . 9 (((𝐸:dom 𝐸1-1→ran 𝐸𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝐴 = 𝐶))
254, 24syld 46 . . . . . . . 8 (((𝐸:dom 𝐸1-1→ran 𝐸𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) = (𝐸‘{𝐵, 𝐶}) → 𝐴 = 𝐶))
2625necon3d 2803 . . . . . . 7 (((𝐸:dom 𝐸1-1→ran 𝐸𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴𝐶 → (𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶})))
2726exp31 628 . . . . . 6 (𝐸:dom 𝐸1-1→ran 𝐸 → (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐴𝐶 → (𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶})))))
2827com34 89 . . . . 5 (𝐸:dom 𝐸1-1→ran 𝐸 → (𝑉 USGrph 𝐸 → (𝐴𝐶 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶})))))
292, 28syl 17 . . . 4 (𝐸:dom 𝐸1-1-onto→ran 𝐸 → (𝑉 USGrph 𝐸 → (𝐴𝐶 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶})))))
301, 29mpcom 37 . . 3 (𝑉 USGrph 𝐸 → (𝐴𝐶 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}))))
3130imp31 447 . 2 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}))
32 usgra2adedgspthlem1 26139 . . 3 ((𝑉 USGrph 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
3332adantlr 747 . 2 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
34 3anass 1035 . 2 (((𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}) ∧ (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}) ↔ ((𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}) ∧ ((𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶})))
3531, 33, 34sylanbrc 695 1 (((𝑉 USGrph 𝐸𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐸‘{𝐴, 𝐵}) ≠ (𝐸‘{𝐵, 𝐶}) ∧ (𝐸‘(𝐸‘{𝐴, 𝐵})) = {𝐴, 𝐵} ∧ (𝐸‘(𝐸‘{𝐵, 𝐶})) = {𝐵, 𝐶}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {cpr 4127   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  ran crn 5039  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804   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-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-int 4411  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-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-usgra 25862 This theorem is referenced by:  usgra2adedgspth  26141
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