Theorem usgredg2vlem2 29303

Description: Lemma 2 for usgredg2v 29304. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)

Hypotheses

Ref	Expression
usgredg2v.v	⊢ 𝑉 = (Vtx‘𝐺)
usgredg2v.e	⊢ 𝐸 = (iEdg‘𝐺)
usgredg2v.a	⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}

Assertion

Ref	Expression
usgredg2vlem2	⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁}))

Distinct variable groups:   𝑥,𝐸,𝑧   𝑧,𝐺   𝑥,𝑁,𝑧   𝑧,𝑉   𝑥,𝑌,𝑧   𝑧,𝐼

Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐺(𝑥)   𝐼(𝑥)   𝑉(𝑥)

Proof of Theorem usgredg2vlem2

Step	Hyp	Ref	Expression
1		fveq2 6835	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐸‘𝑥) = (𝐸‘𝑌))
2	1	eleq2d 2823	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝑁 ∈ (𝐸‘𝑥) ↔ 𝑁 ∈ (𝐸‘𝑌)))
3		usgredg2v.a	. . . . 5 ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}
4	2, 3	elrab2 3650	. . . 4 ⊢ (𝑌 ∈ 𝐴 ↔ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))
5	4	biimpi 216	. . 3 ⊢ (𝑌 ∈ 𝐴 → (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))
6		usgredg2v.v	. . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺)
7		usgredg2v.e	. . . . . . . 8 ⊢ 𝐸 = (iEdg‘𝐺)
8	6, 7	usgredgreu 29295	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)) → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧})
9	8	3expb 1121	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌))) → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧})
10	6, 7, 3	usgredg2vlem1 29302	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
11	10	adantlr 716	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌))) ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
12	11	ad4ant23 754	. . . . . . . . . . . . 13 ⊢ ((((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) ∧ 𝑌 ∈ 𝐴) ∧ 𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
13		eleq1 2825	. . . . . . . . . . . . . 14 ⊢ (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐼 ∈ 𝑉 ↔ (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉))
14	13	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) ∧ 𝑌 ∈ 𝐴) ∧ 𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})) → (𝐼 ∈ 𝑉 ↔ (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉))
15	12, 14	mpbird 257	. . . . . . . . . . . 12 ⊢ ((((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) ∧ 𝑌 ∈ 𝐴) ∧ 𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})) → 𝐼 ∈ 𝑉)
16		prcom 4690	. . . . . . . . . . . . . . . 16 ⊢ {𝑁, 𝑧} = {𝑧, 𝑁}
17	16	eqeq2i 2750	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑌) = {𝑁, 𝑧} ↔ (𝐸‘𝑌) = {𝑧, 𝑁})
18	17	reubii 3360	. . . . . . . . . . . . . 14 ⊢ (∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ↔ ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})
19	18	biimpi 216	. . . . . . . . . . . . 13 ⊢ (∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})
20	19	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) ∧ 𝑌 ∈ 𝐴) ∧ 𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})) → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})
21		preq1 4691	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐼 → {𝑧, 𝑁} = {𝐼, 𝑁})
22	21	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐼 → ((𝐸‘𝑌) = {𝑧, 𝑁} ↔ (𝐸‘𝑌) = {𝐼, 𝑁}))
23	22	riota2 7342	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → ((𝐸‘𝑌) = {𝐼, 𝑁} ↔ (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) = 𝐼))
24	15, 20, 23	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) ∧ 𝑌 ∈ 𝐴) ∧ 𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})) → ((𝐸‘𝑌) = {𝐼, 𝑁} ↔ (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) = 𝐼))
25	24	exbiri 811	. . . . . . . . . 10 ⊢ (((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) ∧ 𝑌 ∈ 𝐴) → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → ((℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) = 𝐼 → (𝐸‘𝑌) = {𝐼, 𝑁})))
26	25	com13 88	. . . . . . . . 9 ⊢ ((℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) = 𝐼 → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) ∧ 𝑌 ∈ 𝐴) → (𝐸‘𝑌) = {𝐼, 𝑁})))
27	26	eqcoms 2745	. . . . . . . 8 ⊢ (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) ∧ 𝑌 ∈ 𝐴) → (𝐸‘𝑌) = {𝐼, 𝑁})))
28	27	pm2.43i 52	. . . . . . 7 ⊢ (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) ∧ 𝑌 ∈ 𝐴) → (𝐸‘𝑌) = {𝐼, 𝑁}))
29	28	expdcom 414	. . . . . 6 ⊢ ((∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))) → (𝑌 ∈ 𝐴 → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁})))
30	9, 29	mpancom 689	. . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌))) → (𝑌 ∈ 𝐴 → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁})))
31	30	expcom 413	. . . 4 ⊢ ((𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)) → (𝐺 ∈ USGraph → (𝑌 ∈ 𝐴 → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁}))))
32	31	com23 86	. . 3 ⊢ ((𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)) → (𝑌 ∈ 𝐴 → (𝐺 ∈ USGraph → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁}))))
33	5, 32	mpcom 38	. 2 ⊢ (𝑌 ∈ 𝐴 → (𝐺 ∈ USGraph → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁})))
34	33	impcom 407	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃!wreu 3349  {crab 3400  {cpr 4583  dom cdm 5625  ‘cfv 6493  ℩crio 7316  Vtxcvtx 29073  iEdgciedg 29074  USGraphcusgr 29226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-cnex 11086  ax-resscn 11087  ax-1cn 11088  ax-icn 11089  ax-addcl 11090  ax-addrcl 11091  ax-mulcl 11092  ax-mulrcl 11093  ax-mulcom 11094  ax-addass 11095  ax-mulass 11096  ax-distr 11097  ax-i2m1 11098  ax-1ne0 11099  ax-1rid 11100  ax-rnegex 11101  ax-rrecex 11102  ax-cnre 11103  ax-pre-lttri 11104  ax-pre-lttrn 11105  ax-pre-ltadd 11106  ax-pre-mulgt0 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9817  df-card 9855  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12150  df-2 12212  df-n0 12406  df-z 12493  df-uz 12756  df-fz 13428  df-hash 14258  df-edg 29125  df-umgr 29160  df-usgr 29228

This theorem is referenced by:  usgredg2v  29304