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Theorem usgredg2vlem2 26924
Description: Lemma 2 for usgredg2v 26925. (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 𝐸𝑁 ∈ (𝐸𝑥)}
Assertion
Ref Expression
usgredg2vlem2 ((𝐺 ∈ USGraph ∧ 𝑌𝐴) → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {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
StepHypRef Expression
1 fveq2 6666 . . . . . 6 (𝑥 = 𝑌 → (𝐸𝑥) = (𝐸𝑌))
21eleq2d 2902 . . . . 5 (𝑥 = 𝑌 → (𝑁 ∈ (𝐸𝑥) ↔ 𝑁 ∈ (𝐸𝑌)))
3 usgredg2v.a . . . . 5 𝐴 = {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}
42, 3elrab2 3686 . . . 4 (𝑌𝐴 ↔ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))
54biimpi 217 . . 3 (𝑌𝐴 → (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))
6 usgredg2v.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
7 usgredg2v.e . . . . . . . 8 𝐸 = (iEdg‘𝐺)
86, 7usgredgreu 26916 . . . . . . 7 ((𝐺 ∈ USGraph ∧ 𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)) → ∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧})
983expb 1114 . . . . . 6 ((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌))) → ∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧})
106, 7, 3usgredg2vlem1 26923 . . . . . . . . . . . . . . 15 ((𝐺 ∈ USGraph ∧ 𝑌𝐴) → (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
1110adantlr 711 . . . . . . . . . . . . . 14 (((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌))) ∧ 𝑌𝐴) → (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
1211ad4ant23 749 . . . . . . . . . . . . 13 ((((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) ∧ 𝑌𝐴) ∧ 𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁})) → (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) ∈ 𝑉)
13 eleq1 2904 . . . . . . . . . . . . . 14 (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐼𝑉 ↔ (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) ∈ 𝑉))
1413adantl 482 . . . . . . . . . . . . 13 ((((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) ∧ 𝑌𝐴) ∧ 𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁})) → (𝐼𝑉 ↔ (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) ∈ 𝑉))
1512, 14mpbird 258 . . . . . . . . . . . 12 ((((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) ∧ 𝑌𝐴) ∧ 𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁})) → 𝐼𝑉)
16 prcom 4666 . . . . . . . . . . . . . . . 16 {𝑁, 𝑧} = {𝑧, 𝑁}
1716eqeq2i 2838 . . . . . . . . . . . . . . 15 ((𝐸𝑌) = {𝑁, 𝑧} ↔ (𝐸𝑌) = {𝑧, 𝑁})
1817reubii 3396 . . . . . . . . . . . . . 14 (∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ↔ ∃!𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁})
1918biimpi 217 . . . . . . . . . . . . 13 (∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} → ∃!𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁})
2019ad3antrrr 726 . . . . . . . . . . . 12 ((((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) ∧ 𝑌𝐴) ∧ 𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁})) → ∃!𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁})
21 preq1 4667 . . . . . . . . . . . . . 14 (𝑧 = 𝐼 → {𝑧, 𝑁} = {𝐼, 𝑁})
2221eqeq2d 2836 . . . . . . . . . . . . 13 (𝑧 = 𝐼 → ((𝐸𝑌) = {𝑧, 𝑁} ↔ (𝐸𝑌) = {𝐼, 𝑁}))
2322riota2 7134 . . . . . . . . . . . 12 ((𝐼𝑉 ∧ ∃!𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → ((𝐸𝑌) = {𝐼, 𝑁} ↔ (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) = 𝐼))
2415, 20, 23syl2anc 584 . . . . . . . . . . 11 ((((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) ∧ 𝑌𝐴) ∧ 𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁})) → ((𝐸𝑌) = {𝐼, 𝑁} ↔ (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) = 𝐼))
2524exbiri 807 . . . . . . . . . 10 (((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) ∧ 𝑌𝐴) → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → ((𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) = 𝐼 → (𝐸𝑌) = {𝐼, 𝑁})))
2625com13 88 . . . . . . . . 9 ((𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) = 𝐼 → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) ∧ 𝑌𝐴) → (𝐸𝑌) = {𝐼, 𝑁})))
2726eqcoms 2833 . . . . . . . 8 (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) ∧ 𝑌𝐴) → (𝐸𝑌) = {𝐼, 𝑁})))
2827pm2.43i 52 . . . . . . 7 (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) ∧ 𝑌𝐴) → (𝐸𝑌) = {𝐼, 𝑁}))
2928expdcom 415 . . . . . 6 ((∃!𝑧𝑉 (𝐸𝑌) = {𝑁, 𝑧} ∧ (𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)))) → (𝑌𝐴 → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {𝐼, 𝑁})))
309, 29mpancom 684 . . . . 5 ((𝐺 ∈ USGraph ∧ (𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌))) → (𝑌𝐴 → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {𝐼, 𝑁})))
3130expcom 414 . . . 4 ((𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)) → (𝐺 ∈ USGraph → (𝑌𝐴 → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {𝐼, 𝑁}))))
3231com23 86 . . 3 ((𝑌 ∈ dom 𝐸𝑁 ∈ (𝐸𝑌)) → (𝑌𝐴 → (𝐺 ∈ USGraph → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {𝐼, 𝑁}))))
335, 32mpcom 38 . 2 (𝑌𝐴 → (𝐺 ∈ USGraph → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {𝐼, 𝑁})))
3433impcom 408 1 ((𝐺 ∈ USGraph ∧ 𝑌𝐴) → (𝐼 = (𝑧𝑉 (𝐸𝑌) = {𝑧, 𝑁}) → (𝐸𝑌) = {𝐼, 𝑁}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  ∃!wreu 3144  {crab 3146  {cpr 4565  dom cdm 5553  cfv 6351  crio 7108  Vtxcvtx 26697  iEdgciedg 26698  USGraphcusgr 26850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-hash 13684  df-edg 26749  df-umgr 26784  df-usgr 26852
This theorem is referenced by:  usgredg2v  26925
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