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 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑖)}
nbgraf1o.f	⊢ 𝐹 = (𝑛 ∈ 𝑁 ↦ (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}))

Assertion

Ref	Expression
nbgraf1olem3	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {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

Step	Hyp	Ref	Expression
1		nbgraf1o.n	. . . . 5 ⊢ 𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑈)
2	1	eleq2i 2680	. . . 4 ⊢ (𝑀 ∈ 𝑁 ↔ 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈))
3		nbgracnvfv 25969	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈)) → (𝐸‘(◡𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀})
4	2, 3	sylan2b 491	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑀 ∈ 𝑁) → (𝐸‘(◡𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀})
5	4	3adant2 1073	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (𝐸‘(◡𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀})
6		usgraf1o 25887	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → 𝐸:dom 𝐸–1-1-onto→ran 𝐸)
7	6	3ad2ant1 1075	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → 𝐸:dom 𝐸–1-1-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}
10	9	eleq1i 2679	. . . . . . . . . . . 12 ⊢ ({𝑀, 𝑈} ∈ ran 𝐸 ↔ {𝑈, 𝑀} ∈ ran 𝐸)
11	8, 10	syl6bb 275	. . . . . . . . . . 11 ⊢ (𝑉 USGrph 𝐸 → (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) ↔ {𝑈, 𝑀} ∈ ran 𝐸))
12	11	biimpcd 238	. . . . . . . . . 10 ⊢ (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → (𝑉 USGrph 𝐸 → {𝑈, 𝑀} ∈ ran 𝐸))
13	12	a1d 25	. . . . . . . . 9 ⊢ (𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑈) → (𝑈 ∈ 𝑉 → (𝑉 USGrph 𝐸 → {𝑈, 𝑀} ∈ ran 𝐸)))
14	13, 1	eleq2s 2706	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑁 → (𝑈 ∈ 𝑉 → (𝑉 USGrph 𝐸 → {𝑈, 𝑀} ∈ ran 𝐸)))
15	14	com13 86	. . . . . . 7 ⊢ (𝑉 USGrph 𝐸 → (𝑈 ∈ 𝑉 → (𝑀 ∈ 𝑁 → {𝑈, 𝑀} ∈ ran 𝐸)))
16	15	3imp 1249	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → {𝑈, 𝑀} ∈ ran 𝐸)
17		f1ocnvdm 6440	. . . . . 6 ⊢ ((𝐸:dom 𝐸–1-1-onto→ran 𝐸 ∧ {𝑈, 𝑀} ∈ ran 𝐸) → (◡𝐸‘{𝑈, 𝑀}) ∈ dom 𝐸)
18	7, 16, 17	syl2anc 691	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (◡𝐸‘{𝑈, 𝑀}) ∈ dom 𝐸)
19		prid1g 4239	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ {𝑈, 𝑀})
20		eleq2 2677	. . . . . . . 8 ⊢ ((𝐸‘(◡𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} → (𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑀})) ↔ 𝑈 ∈ {𝑈, 𝑀}))
21	19, 20	syl5ibrcom 236	. . . . . . 7 ⊢ (𝑈 ∈ 𝑉 → ((𝐸‘(◡𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} → 𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑀}))))
22	21	3ad2ant2 1076	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → ((𝐸‘(◡𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} → 𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑀}))))
23	5, 22	mpd 15	. . . . 5 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → 𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑀})))
24		fveq2 6103	. . . . . . 7 ⊢ (𝑖 = (◡𝐸‘{𝑈, 𝑀}) → (𝐸‘𝑖) = (𝐸‘(◡𝐸‘{𝑈, 𝑀})))
25	24	eleq2d 2673	. . . . . 6 ⊢ (𝑖 = (◡𝐸‘{𝑈, 𝑀}) → (𝑈 ∈ (𝐸‘𝑖) ↔ 𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑀}))))
26	25	elrab 3331	. . . . 5 ⊢ ((◡𝐸‘{𝑈, 𝑀}) ∈ {𝑖 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑖)} ↔ ((◡𝐸‘{𝑈, 𝑀}) ∈ dom 𝐸 ∧ 𝑈 ∈ (𝐸‘(◡𝐸‘{𝑈, 𝑀}))))
27	18, 23, 26	sylanbrc 695	. . . 4 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (◡𝐸‘{𝑈, 𝑀}) ∈ {𝑖 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑖)})
28		nbgraf1o.i	. . . 4 ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑖)}
29	27, 28	syl6eleqr 2699	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (◡𝐸‘{𝑈, 𝑀}) ∈ 𝐼)
30		nbgraf1o.f	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ 𝑁 ↦ (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑛}))
31	1, 28, 30	nbgraf1olem1 25970	. . . 4 ⊢ (((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) ∧ 𝑀 ∈ 𝑁) → ∃!𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑀})
32	31	3impa 1251	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → ∃!𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑀})
33	24	eqeq1d 2612	. . . 4 ⊢ (𝑖 = (◡𝐸‘{𝑈, 𝑀}) → ((𝐸‘𝑖) = {𝑈, 𝑀} ↔ (𝐸‘(◡𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀}))
34	33	riota2 6533	. . 3 ⊢ (((◡𝐸‘{𝑈, 𝑀}) ∈ 𝐼 ∧ ∃!𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑀}) → ((𝐸‘(◡𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} ↔ (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑀}) = (◡𝐸‘{𝑈, 𝑀})))
35	29, 32, 34	syl2anc 691	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → ((𝐸‘(◡𝐸‘{𝑈, 𝑀})) = {𝑈, 𝑀} ↔ (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑀}) = (◡𝐸‘{𝑈, 𝑀})))
36	5, 35	mpbid 221	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝑀 ∈ 𝑁) → (℩𝑖 ∈ 𝐼 (𝐸‘𝑖) = {𝑈, 𝑀}) = (◡𝐸‘{𝑈, 𝑀}))

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