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Theorem nbgraf1olem3 25972
 Description: Lemma 3 for nbgraf1o 25976. The restricted iota of an edge is the function value of the converse applied to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
Hypotheses
Ref Expression
nbgraf1o.n 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)
nbgraf1o.i 𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}
nbgraf1o.f 𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))
Assertion
Ref Expression
nbgraf1olem3 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀}) = (𝐸‘{𝑈, 𝑀}))
Distinct variable groups:   𝑖,𝐸,𝑛   𝑈,𝑖,𝑛   𝑖,𝑉,𝑛   𝑖,𝐼,𝑛   𝑛,𝑁   𝑖,𝑀
Allowed substitution hints:   𝐹(𝑖,𝑛)   𝑀(𝑛)   𝑁(𝑖)

Proof of Theorem nbgraf1olem3
StepHypRef Expression
1 nbgraf1o.n . . . . 5 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)
21eleq2i 2680 . . . 4 (𝑀𝑁𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
3 nbgracnvfv 25969 . . . 4 ((𝑉 USGrph 𝐸𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈)) → (𝐸‘(𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀})
42, 3sylan2b 491 . . 3 ((𝑉 USGrph 𝐸𝑀𝑁) → (𝐸‘(𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀})
543adant2 1073 . 2 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝐸‘(𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀})
6 usgraf1o 25887 . . . . . . 7 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)
763ad2ant1 1075 . . . . . 6 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → 𝐸:dom 𝐸1-1-onto→ran 𝐸)
8 nbgraeledg 25959 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸 → (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑀, 𝑈} ∈ ran 𝐸))
9 prcom 4211 . . . . . . . . . . . . 13 {𝑀, 𝑈} = {𝑈, 𝑀}
109eleq1i 2679 . . . . . . . . . . . 12 ({𝑀, 𝑈} ∈ ran 𝐸 ↔ {𝑈, 𝑀} ∈ ran 𝐸)
118, 10syl6bb 275 . . . . . . . . . . 11 (𝑉 USGrph 𝐸 → (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑈, 𝑀} ∈ ran 𝐸))
1211biimpcd 238 . . . . . . . . . 10 (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → (𝑉 USGrph 𝐸 → {𝑈, 𝑀} ∈ ran 𝐸))
1312a1d 25 . . . . . . . . 9 (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → (𝑈𝑉 → (𝑉 USGrph 𝐸 → {𝑈, 𝑀} ∈ ran 𝐸)))
1413, 1eleq2s 2706 . . . . . . . 8 (𝑀𝑁 → (𝑈𝑉 → (𝑉 USGrph 𝐸 → {𝑈, 𝑀} ∈ ran 𝐸)))
1514com13 86 . . . . . . 7 (𝑉 USGrph 𝐸 → (𝑈𝑉 → (𝑀𝑁 → {𝑈, 𝑀} ∈ ran 𝐸)))
16153imp 1249 . . . . . 6 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → {𝑈, 𝑀} ∈ ran 𝐸)
17 f1ocnvdm 6440 . . . . . 6 ((𝐸:dom 𝐸1-1-onto→ran 𝐸 ∧ {𝑈, 𝑀} ∈ ran 𝐸) → (𝐸‘{𝑈, 𝑀}) ∈ dom 𝐸)
187, 16, 17syl2anc 691 . . . . 5 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝐸‘{𝑈, 𝑀}) ∈ dom 𝐸)
19 prid1g 4239 . . . . . . . 8 (𝑈𝑉𝑈 ∈ {𝑈, 𝑀})
20 eleq2 2677 . . . . . . . 8 ((𝐸‘(𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} → (𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑀})) ↔ 𝑈 ∈ {𝑈, 𝑀}))
2119, 20syl5ibrcom 236 . . . . . . 7 (𝑈𝑉 → ((𝐸‘(𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} → 𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑀}))))
22213ad2ant2 1076 . . . . . 6 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → ((𝐸‘(𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} → 𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑀}))))
235, 22mpd 15 . . . . 5 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → 𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑀})))
24 fveq2 6103 . . . . . . 7 (𝑖 = (𝐸‘{𝑈, 𝑀}) → (𝐸𝑖) = (𝐸‘(𝐸‘{𝑈, 𝑀})))
2524eleq2d 2673 . . . . . 6 (𝑖 = (𝐸‘{𝑈, 𝑀}) → (𝑈 ∈ (𝐸𝑖) ↔ 𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑀}))))
2625elrab 3331 . . . . 5 ((𝐸‘{𝑈, 𝑀}) ∈ {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)} ↔ ((𝐸‘{𝑈, 𝑀}) ∈ dom 𝐸𝑈 ∈ (𝐸‘(𝐸‘{𝑈, 𝑀}))))
2718, 23, 26sylanbrc 695 . . . 4 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝐸‘{𝑈, 𝑀}) ∈ {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)})
28 nbgraf1o.i . . . 4 𝐼 = {𝑖 ∈ dom 𝐸𝑈 ∈ (𝐸𝑖)}
2927, 28syl6eleqr 2699 . . 3 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝐸‘{𝑈, 𝑀}) ∈ 𝐼)
30 nbgraf1o.f . . . . 5 𝐹 = (𝑛𝑁 ↦ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑛}))
311, 28, 30nbgraf1olem1 25970 . . . 4 (((𝑉 USGrph 𝐸𝑈𝑉) ∧ 𝑀𝑁) → ∃!𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀})
32313impa 1251 . . 3 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → ∃!𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀})
3324eqeq1d 2612 . . . 4 (𝑖 = (𝐸‘{𝑈, 𝑀}) → ((𝐸𝑖) = {𝑈, 𝑀} ↔ (𝐸‘(𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀}))
3433riota2 6533 . . 3 (((𝐸‘{𝑈, 𝑀}) ∈ 𝐼 ∧ ∃!𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀}) → ((𝐸‘(𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} ↔ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀}) = (𝐸‘{𝑈, 𝑀})))
3529, 32, 34syl2anc 691 . 2 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → ((𝐸‘(𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} ↔ (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀}) = (𝐸‘{𝑈, 𝑀})))
365, 35mpbid 221 1 ((𝑉 USGrph 𝐸𝑈𝑉𝑀𝑁) → (𝑖𝐼 (𝐸𝑖) = {𝑈, 𝑀}) = (𝐸‘{𝑈, 𝑀}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃!wreu 2898  {crab 2900  {cpr 4127  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  ran crn 5039  –1-1-onto→wf1o 5803  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   USGrph cusg 25859   Neighbors cnbgra 25946 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-rmo 2904  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  df-nbgra 25949 This theorem is referenced by:  nbgraf1olem4  25973  nbgraf1olem5  25974
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