Theorem usgredg2vlem2 29283

Description: Lemma 2 for usgredg2v 29284. (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 6829	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐸‘𝑥) = (𝐸‘𝑌))
2	1	eleq2d 2821	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝑁 ∈ (𝐸‘𝑥) ↔ 𝑁 ∈ (𝐸‘𝑌)))
3		usgredg2v.a	. . . . 5 ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}
4	2, 3	elrab2 3634	. . . 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 29275	. . . . . . 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 29282	. . . . . . . . . . . . . . 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 2823	. . . . . . . . . . . . . 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 4666	. . . . . . . . . . . . . . . 16 ⊢ {𝑁, 𝑧} = {𝑧, 𝑁}
17	16	eqeq2i 2748	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑌) = {𝑁, 𝑧} ↔ (𝐸‘𝑌) = {𝑧, 𝑁})
18	17	reubii 3349	. . . . . . . . . . . . . 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 4667	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐼 → {𝑧, 𝑁} = {𝐼, 𝑁})
22	21	eqeq2d 2746	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐼 → ((𝐸‘𝑌) = {𝑧, 𝑁} ↔ (𝐸‘𝑌) = {𝐼, 𝑁}))
23	22	riota2 7338	. . . . . . . . . . . 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 2743	. . . . . . . 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 3338  {crab 3387  {cpr 4559  dom cdm 5620  ‘cfv 6487  ℩crio 7312  Vtxcvtx 29053  iEdgciedg 29054  USGraphcusgr 29206

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 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-oadd 8398  df-er 8632  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-dju 9814  df-card 9852  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-n0 12427  df-z 12514  df-uz 12778  df-fz 13451  df-hash 14282  df-edg 29105  df-umgr 29140  df-usgr 29208

This theorem is referenced by:  usgredg2v  29284