Theorem usgredg2vlem2 29052

Description: Lemma 2 for usgredg2v 29053. (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 6897	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐸‘𝑥) = (𝐸‘𝑌))
2	1	eleq2d 2815	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝑁 ∈ (𝐸‘𝑥) ↔ 𝑁 ∈ (𝐸‘𝑌)))
3		usgredg2v.a	. . . . 5 ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}
4	2, 3	elrab2 3685	. . . 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 29044	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)) → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧})
9	8	3expb 1118	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌))) → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧})
10	6, 7, 3	usgredg2vlem1 29051	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
11	10	adantlr 714	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌))) ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
12	11	ad4ant23 752	. . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . 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 4737	. . . . . . . . . . . . . . . 16 ⊢ {𝑁, 𝑧} = {𝑧, 𝑁}
17	16	eqeq2i 2741	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑌) = {𝑁, 𝑧} ↔ (𝐸‘𝑌) = {𝑧, 𝑁})
18	17	reubii 3382	. . . . . . . . . . . . . 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 729	. . . . . . . . . . . 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 4738	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐼 → {𝑧, 𝑁} = {𝐼, 𝑁})
22	21	eqeq2d 2739	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐼 → ((𝐸‘𝑌) = {𝑧, 𝑁} ↔ (𝐸‘𝑌) = {𝐼, 𝑁}))
23	22	riota2 7402	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 810	. . . . . . . . . 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 2736	. . . . . . . 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 687	. . . . 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 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∃!wreu 3371  {crab 3429  {cpr 4631  dom cdm 5678  ‘cfv 6548  ℩crio 7375  Vtxcvtx 28822  iEdgciedg 28823  USGraphcusgr 28975

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-dju 9925  df-card 9963  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-2 12306  df-n0 12504  df-z 12590  df-uz 12854  df-fz 13518  df-hash 14323  df-edg 28874  df-umgr 28909  df-usgr 28977

This theorem is referenced by:  usgredg2v  29053