Theorem usgredg2vlem2 26118

Description: Lemma 2 for usgredg2v 26119. (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 6191	. . . . . 6 ⊢ (𝑥 = 𝑌 → (𝐸‘𝑥) = (𝐸‘𝑌))
2	1	eleq2d 2687	. . . . 5 ⊢ (𝑥 = 𝑌 → (𝑁 ∈ (𝐸‘𝑥) ↔ 𝑁 ∈ (𝐸‘𝑌)))
3		usgredg2v.a	. . . . 5 ⊢ 𝐴 = {𝑥 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑥)}
4	2, 3	elrab2 3366	. . . 4 ⊢ (𝑌 ∈ 𝐴 ↔ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)))
5	4	biimpi 206	. . 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 26110	. . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌)) → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧})
9	8	3expb 1266	. . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌))) → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧})
10	6, 7, 3	usgredg2vlem1 26117	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
11	10	adantlr 751	. . . . . . . . . . . . . 14 ⊢ (((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸 ∧ 𝑁 ∈ (𝐸‘𝑌))) ∧ 𝑌 ∈ 𝐴) → (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
12	11	ad4ant23 1297	. . . . . . . . . . . . 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 2689	. . . . . . . . . . . . . 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 247	. . . . . . . . . . . 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 4267	. . . . . . . . . . . . . . . 16 ⊢ {𝑁, 𝑧} = {𝑧, 𝑁}
17	16	eqeq2i 2634	. . . . . . . . . . . . . . 15 ⊢ ((𝐸‘𝑌) = {𝑁, 𝑧} ↔ (𝐸‘𝑌) = {𝑧, 𝑁})
18	17	reubii 3128	. . . . . . . . . . . . . 14 ⊢ (∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} ↔ ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})
19	18	biimpi 206	. . . . . . . . . . . . 13 ⊢ (∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑁, 𝑧} → ∃!𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁})
20	19	ad3antrrr 766	. . . . . . . . . . . 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 4268	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝐼 → {𝑧, 𝑁} = {𝐼, 𝑁})
22	21	eqeq2d 2632	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐼 → ((𝐸‘𝑌) = {𝑧, 𝑁} ↔ (𝐸‘𝑌) = {𝐼, 𝑁}))
23	22	riota2 6633	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 652	. . . . . . . . . 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 2630	. . . . . . . 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 455	. . . . . 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 703	. . . . 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 451	. . . 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 446	1 ⊢ ((𝐺 ∈ USGraph ∧ 𝑌 ∈ 𝐴) → (𝐼 = (℩𝑧 ∈ 𝑉 (𝐸‘𝑌) = {𝑧, 𝑁}) → (𝐸‘𝑌) = {𝐼, 𝑁}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃!wreu 2914  {crab 2916  {cpr 4179  dom cdm 5114  ‘cfv 5888  ℩crio 6610  Vtxcvtx 25874  iEdgciedg 25875   USGraph cusgr 26044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-edg 25940  df-umgr 25978  df-usgr 26046

This theorem is referenced by:  usgredg2v  26119