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Theorem uspgrsprf 42079
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 2722 . . . 4 (𝑔𝐺𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
4 elopab 5012 . . . 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 264 . . 3 (𝑔𝐺 ↔ ∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
6 uspgrupgr 26116 . . . . . . . . . . . . 13 (𝑞 ∈ USPGraph → 𝑞 ∈ UPGraph)
7 upgredgssspr 42076 . . . . . . . . . . . . 13 (𝑞 ∈ UPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
86, 7syl 17 . . . . . . . . . . . 12 (𝑞 ∈ USPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
98adantr 480 . . . . . . . . . . 11 ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
10 simpr 476 . . . . . . . . . . . . 13 (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → (Edg‘𝑞) = 𝑒)
11 fveq2 6229 . . . . . . . . . . . . . 14 ((Vtx‘𝑞) = 𝑣 → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
1211adantr 480 . . . . . . . . . . . . 13 (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
1310, 12sseq12d 3667 . . . . . . . . . . . 12 (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → ((Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)) ↔ 𝑒 ⊆ (Pairs‘𝑣)))
1413adantl 481 . . . . . . . . . . 11 ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → ((Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)) ↔ 𝑒 ⊆ (Pairs‘𝑣)))
159, 14mpbid 222 . . . . . . . . . 10 ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑣))
1615rexlimiva 3057 . . . . . . . . 9 (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → 𝑒 ⊆ (Pairs‘𝑣))
1716adantl 481 . . . . . . . 8 ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑣))
18 fveq2 6229 . . . . . . . . . 10 (𝑣 = 𝑉 → (Pairs‘𝑣) = (Pairs‘𝑉))
1918sseq2d 3666 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑒 ⊆ (Pairs‘𝑣) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2019adantr 480 . . . . . . . 8 ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → (𝑒 ⊆ (Pairs‘𝑣) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2117, 20mpbid 222 . . . . . . 7 ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑉))
2221adantl 481 . . . . . 6 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → 𝑒 ⊆ (Pairs‘𝑉))
23 vex 3234 . . . . . . . . 9 𝑣 ∈ V
24 vex 3234 . . . . . . . . 9 𝑒 ∈ V
2523, 24op2ndd 7221 . . . . . . . 8 (𝑔 = ⟨𝑣, 𝑒⟩ → (2nd𝑔) = 𝑒)
2625sseq1d 3665 . . . . . . 7 (𝑔 = ⟨𝑣, 𝑒⟩ → ((2nd𝑔) ⊆ (Pairs‘𝑉) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2726adantr 480 . . . . . 6 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((2nd𝑔) ⊆ (Pairs‘𝑉) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2822, 27mpbird 247 . . . . 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 2722 . . . . . 6 ((2nd𝑔) ∈ 𝑃 ↔ (2nd𝑔) ∈ 𝒫 (Pairs‘𝑉))
31 fvex 6239 . . . . . . 7 (2nd𝑔) ∈ V
3231elpw 4197 . . . . . 6 ((2nd𝑔) ∈ 𝒫 (Pairs‘𝑉) ↔ (2nd𝑔) ⊆ (Pairs‘𝑉))
3330, 32bitri 264 . . . . 5 ((2nd𝑔) ∈ 𝑃 ↔ (2nd𝑔) ⊆ (Pairs‘𝑉))
3428, 33sylibr 224 . . . 4 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd𝑔) ∈ 𝑃)
3534exlimivv 1900 . . 3 (∃𝑣𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd𝑔) ∈ 𝑃)
365, 35sylbi 207 . 2 (𝑔𝐺 → (2nd𝑔) ∈ 𝑃)
371, 36fmpti 6423 1 𝐹:𝐺𝑃
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  wss 3607  𝒫 cpw 4191  cop 4216  {copab 4745  cmpt 4762  wf 5922  cfv 5926  2nd c2nd 7209  Vtxcvtx 25919  Edgcedg 25984  UPGraphcupgr 26020  USPGraphcuspgr 26088  Pairscspr 42052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-hash 13158  df-edg 25985  df-upgr 26022  df-uspgr 26090  df-spr 42053
This theorem is referenced by:  uspgrsprf1  42080  uspgrsprfo  42081
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