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Theorem uspgrsprf 43872
Description: The mapping 𝐹 is a function from the "simple pseudographs" with a fixed set of vertices 𝑉 into the subsets of the set of pairs over the set 𝑉. (Contributed by AV, 24-Nov-2021.)
Hypotheses
Ref Expression
uspgrsprf.p 𝑃 = 𝒫 (Pairs‘𝑉)
uspgrsprf.g 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
uspgrsprf.f 𝐹 = (𝑔𝐺 ↦ (2nd𝑔))
Assertion
Ref Expression
uspgrsprf 𝐹:𝐺𝑃
Distinct variable groups:   𝑃,𝑒,𝑞,𝑣   𝑒,𝑉,𝑞,𝑣   𝑔,𝐺   𝑃,𝑔,𝑒,𝑣
Allowed substitution hints:   𝐹(𝑣,𝑒,𝑔,𝑞)   𝐺(𝑣,𝑒,𝑞)   𝑉(𝑔)

Proof of Theorem uspgrsprf
StepHypRef Expression
1 uspgrsprf.f . 2 𝐹 = (𝑔𝐺 ↦ (2nd𝑔))
2 uspgrsprf.g . . . . 5 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
32eleq2i 2909 . . . 4 (𝑔𝐺𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
4 elopab 5411 . . . 4 (𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ ∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
53, 4bitri 276 . . 3 (𝑔𝐺 ↔ ∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
6 uspgrupgr 26894 . . . . . . . . . . . . 13 (𝑞 ∈ USPGraph → 𝑞 ∈ UPGraph)
7 upgredgssspr 43869 . . . . . . . . . . . . 13 (𝑞 ∈ UPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
86, 7syl 17 . . . . . . . . . . . 12 (𝑞 ∈ USPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
98adantr 481 . . . . . . . . . . 11 ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
10 simpr 485 . . . . . . . . . . . . 13 (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → (Edg‘𝑞) = 𝑒)
11 fveq2 6669 . . . . . . . . . . . . . 14 ((Vtx‘𝑞) = 𝑣 → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
1211adantr 481 . . . . . . . . . . . . 13 (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
1310, 12sseq12d 4004 . . . . . . . . . . . 12 (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → ((Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)) ↔ 𝑒 ⊆ (Pairs‘𝑣)))
1413adantl 482 . . . . . . . . . . 11 ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → ((Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)) ↔ 𝑒 ⊆ (Pairs‘𝑣)))
159, 14mpbid 233 . . . . . . . . . 10 ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑣))
1615rexlimiva 3286 . . . . . . . . 9 (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → 𝑒 ⊆ (Pairs‘𝑣))
1716adantl 482 . . . . . . . 8 ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑣))
18 fveq2 6669 . . . . . . . . . 10 (𝑣 = 𝑉 → (Pairs‘𝑣) = (Pairs‘𝑉))
1918sseq2d 4003 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑒 ⊆ (Pairs‘𝑣) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2019adantr 481 . . . . . . . 8 ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → (𝑒 ⊆ (Pairs‘𝑣) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2117, 20mpbid 233 . . . . . . 7 ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑉))
2221adantl 482 . . . . . 6 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → 𝑒 ⊆ (Pairs‘𝑉))
23 vex 3503 . . . . . . . . 9 𝑣 ∈ V
24 vex 3503 . . . . . . . . 9 𝑒 ∈ V
2523, 24op2ndd 7696 . . . . . . . 8 (𝑔 = ⟨𝑣, 𝑒⟩ → (2nd𝑔) = 𝑒)
2625sseq1d 4002 . . . . . . 7 (𝑔 = ⟨𝑣, 𝑒⟩ → ((2nd𝑔) ⊆ (Pairs‘𝑉) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2726adantr 481 . . . . . 6 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((2nd𝑔) ⊆ (Pairs‘𝑉) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2822, 27mpbird 258 . . . . 5 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd𝑔) ⊆ (Pairs‘𝑉))
29 uspgrsprf.p . . . . . . 7 𝑃 = 𝒫 (Pairs‘𝑉)
3029eleq2i 2909 . . . . . 6 ((2nd𝑔) ∈ 𝑃 ↔ (2nd𝑔) ∈ 𝒫 (Pairs‘𝑉))
31 fvex 6682 . . . . . . 7 (2nd𝑔) ∈ V
3231elpw 4549 . . . . . 6 ((2nd𝑔) ∈ 𝒫 (Pairs‘𝑉) ↔ (2nd𝑔) ⊆ (Pairs‘𝑉))
3330, 32bitri 276 . . . . 5 ((2nd𝑔) ∈ 𝑃 ↔ (2nd𝑔) ⊆ (Pairs‘𝑉))
3428, 33sylibr 235 . . . 4 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd𝑔) ∈ 𝑃)
3534exlimivv 1926 . . 3 (∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd𝑔) ∈ 𝑃)
365, 35sylbi 218 . 2 (𝑔𝐺 → (2nd𝑔) ∈ 𝑃)
371, 36fmpti 6874 1 𝐹:𝐺𝑃
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wex 1773  wcel 2107  wrex 3144  wss 3940  𝒫 cpw 4542  cop 4570  {copab 5125  cmpt 5143  wf 6350  cfv 6354  2nd c2nd 7684  Vtxcvtx 26714  Edgcedg 26765  UPGraphcupgr 26798  USPGraphcuspgr 26866  Pairscspr 43490
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 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
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 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8284  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-dju 9324  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-n0 11892  df-xnn0 11962  df-z 11976  df-uz 12238  df-fz 12888  df-hash 13686  df-edg 26766  df-upgr 26800  df-uspgr 26868  df-spr 43491
This theorem is referenced by:  uspgrsprf1  43873  uspgrsprfo  43874
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