Theorem usgra2adedgspthlem2 26140

Description: Lemma 2 for usgra2adedgspth 26141. (Contributed by Alexander van der Vekens, 1-Feb-2018.)

Assertion

Ref	Expression
usgra2adedgspthlem2	⊢ (((𝑉 USGrph 𝐸 ∧ 𝐴 ≠ 𝐶) ∧ ({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}))

Proof of Theorem usgra2adedgspthlem2

Step	Hyp	Ref	Expression
1		usgraf1o 25887	. . . 4 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
2		f1of1 6049	. . . . 5 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → 𝐸:dom 𝐸–1-1→ran 𝐸)
3		f1ocnvfvrneq 6441	. . . . . . . . . 10 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((◡𝐸‘{𝐴, 𝐵}) = (◡𝐸‘{𝐵, 𝐶}) → {𝐴, 𝐵} = {𝐵, 𝐶}))
4	3	adantlr 747	. . . . . . . . 9 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝑉 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) → (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
6		usgraedgrnv 25906	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))
7		pm3.2 462	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
8	6, 7	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐴, 𝐵} ∈ ran 𝐸) → ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
9	8	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑉 USGrph 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
10	9	com13 86	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
11	5, 10	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑉 USGrph 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
12	11	ex 449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 USGrph 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸 → ({𝐴, 𝐵} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))))
13	12	com13 86	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴, 𝐵} ∈ ran 𝐸 → ({𝐵, 𝐶} ∈ ran 𝐸 → (𝑉 USGrph 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))))
14	13	imp 444	. . . . . . . . . . . . . . 15 ⊢ (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝑉 USGrph 𝐸 → (𝑉 USGrph 𝐸 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
15	14	com13 86	. . . . . . . . . . . . . 14 ⊢ (𝑉 USGrph 𝐸 → (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))))
16	15	pm2.43i 50	. . . . . . . . . . . . 13 ⊢ (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
17	16	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝑉 USGrph 𝐸) → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))))
18	17	imp 444	. . . . . . . . . . 11 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
19		preq12bg 4326	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))))
20	18, 19	syl 17	. . . . . . . . . 10 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝐵} = {𝐵, 𝐶} ↔ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵))))
21		eqtr 2629	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) → 𝐴 = 𝐶)
22		simpl 472	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐵) → 𝐴 = 𝐶)
23	21, 22	jaoi 393	. . . . . . . . . 10 ⊢ (((𝐴 = 𝐵 ∧ 𝐵 = 𝐶) ∨ (𝐴 = 𝐶 ∧ 𝐵 = 𝐵)) → 𝐴 = 𝐶)
24	20, 23	syl6bi 242	. . . . . . . . 9 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ({𝐴, 𝐵} = {𝐵, 𝐶} → 𝐴 = 𝐶))
25	4, 24	syld 46	. . . . . . . 8 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → ((◡𝐸‘{𝐴, 𝐵}) = (◡𝐸‘{𝐵, 𝐶}) → 𝐴 = 𝐶))
26	25	necon3d 2803	. . . . . . 7 ⊢ (((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝑉 USGrph 𝐸) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)) → (𝐴 ≠ 𝐶 → (◡𝐸‘{𝐴, 𝐵}) ≠ (◡𝐸‘{𝐵, 𝐶})))
27	26	exp31 628	. . . . . 6 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → (𝑉 USGrph 𝐸 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (𝐴 ≠ 𝐶 → (◡𝐸‘{𝐴, 𝐵}) ≠ (◡𝐸‘{𝐵, 𝐶})))))
28	27	com34 89	. . . . 5 ⊢ (𝐸:dom 𝐸–1-1→ran 𝐸 → (𝑉 USGrph 𝐸 → (𝐴 ≠ 𝐶 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (◡𝐸‘{𝐴, 𝐵}) ≠ (◡𝐸‘{𝐵, 𝐶})))))
29	2, 28	syl 17	. . . 4 ⊢ (𝐸:dom 𝐸–1-1-onto→ran 𝐸 → (𝑉 USGrph 𝐸 → (𝐴 ≠ 𝐶 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (◡𝐸‘{𝐴, 𝐵}) ≠ (◡𝐸‘{𝐵, 𝐶})))))
30	1, 29	mpcom 37	. . 3 ⊢ (𝑉 USGrph 𝐸 → (𝐴 ≠ 𝐶 → (({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸) → (◡𝐸‘{𝐴, 𝐵}) ≠ (◡𝐸‘{𝐵, 𝐶}))))
31	30	imp31 447	. 2 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝐴 ≠ 𝐶) ∧ ({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}))
33	32	adantlr 747	. 2 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝐴 ≠ 𝐶) ∧ ({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})))
35	31, 33, 34	sylanbrc 695	1 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝐴 ≠ 𝐶) ∧ ({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}))

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