Theorem usgredg2vlem2 28237

Description: Lemma 2 for usgredg2v 28238. (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 6847	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐸‘𝑥) = (𝐸‘𝑌))
2	1	eleq2d 2818	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝑁 ∈ (𝐸‘𝑥) ↔ 𝑁 ∈ (𝐸‘𝑌)))
3		usgredg2v.a	. . . . 5 ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}
4	2, 3	elrab2 3651	. . . 4 ⊢ (𝑌 ∈ 𝐴 ↔ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))
5	4	biimpi 215	. . 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 28229	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)) → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧})
9	8	3expb 1120	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌))) → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧})
10	6, 7, 3	usgredg2vlem1 28236	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
11	10	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌))) ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
12	11	ad4ant23 751	. . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . 14 ⊢ (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐼 ∈ 𝑉 ↔ (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉))
14	13	adantl 482	. . . . . . . . . . . . 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 256	. . . . . . . . . . . 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 4698	. . . . . . . . . . . . . . . 16 ⊢ {𝑁, 𝑧} = {𝑧, 𝑁}
17	16	eqeq2i 2744	. . . . . . . . . . . . . . 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 215	. . . . . . . . . . . . 13 ⊢ (∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})
20	19	ad3antrrr 728	. . . . . . . . . . . 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 4699	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐼 → {𝑧, 𝑁} = {𝐼, 𝑁})
22	21	eqeq2d 2742	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐼 → ((𝐸‘𝑌) = {𝑧, 𝑁} ↔ (𝐸‘𝑌) = {𝐼, 𝑁}))
23	22	riota2 7344	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 809	. . . . . . . . . 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 2739	. . . . . . . 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 415	. . . . . 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 686	. . . . 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 414	. . . 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 408	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃!wreu 3349  {crab 3405  {cpr 4593  dom cdm 5638  ‘cfv 6501  ℩crio 7317  Vtxcvtx 28010  iEdgciedg 28011  USGraphcusgr 28163

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9846  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-n0 12423  df-z 12509  df-uz 12773  df-fz 13435  df-hash 14241  df-edg 28062  df-umgr 28097  df-usgr 28165

This theorem is referenced by:  usgredg2v  28238