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Theorem uspgrsprf 42282
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 2842 . . . 4 (𝑔𝐺𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
4 elopab 5116 . . . 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 26293 . . . . . . . . . . . . 13 (𝑞 ∈ USPGraph → 𝑞 ∈ UPGraph)
7 upgredgssspr 42279 . . . . . . . . . . . . 13 (𝑞 ∈ UPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
86, 7syl 17 . . . . . . . . . . . 12 (𝑞 ∈ USPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
98adantr 466 . . . . . . . . . . 11 ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
10 simpr 471 . . . . . . . . . . . . 13 (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → (Edg‘𝑞) = 𝑒)
11 fveq2 6332 . . . . . . . . . . . . . 14 ((Vtx‘𝑞) = 𝑣 → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
1211adantr 466 . . . . . . . . . . . . 13 (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
1310, 12sseq12d 3783 . . . . . . . . . . . 12 (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → ((Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)) ↔ 𝑒 ⊆ (Pairs‘𝑣)))
1413adantl 467 . . . . . . . . . . 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 3176 . . . . . . . . 9 (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → 𝑒 ⊆ (Pairs‘𝑣))
1716adantl 467 . . . . . . . 8 ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑣))
18 fveq2 6332 . . . . . . . . . 10 (𝑣 = 𝑉 → (Pairs‘𝑣) = (Pairs‘𝑉))
1918sseq2d 3782 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑒 ⊆ (Pairs‘𝑣) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2019adantr 466 . . . . . . . 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 467 . . . . . 6 ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → 𝑒 ⊆ (Pairs‘𝑉))
23 vex 3354 . . . . . . . . 9 𝑣 ∈ V
24 vex 3354 . . . . . . . . 9 𝑒 ∈ V
2523, 24op2ndd 7326 . . . . . . . 8 (𝑔 = ⟨𝑣, 𝑒⟩ → (2nd𝑔) = 𝑒)
2625sseq1d 3781 . . . . . . 7 (𝑔 = ⟨𝑣, 𝑒⟩ → ((2nd𝑔) ⊆ (Pairs‘𝑉) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
2726adantr 466 . . . . . 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 2842 . . . . . 6 ((2nd𝑔) ∈ 𝑃 ↔ (2nd𝑔) ∈ 𝒫 (Pairs‘𝑉))
31 fvex 6342 . . . . . . 7 (2nd𝑔) ∈ V
3231elpw 4303 . . . . . 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 2012 . . 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 6525 1 𝐹:𝐺𝑃
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wrex 3062  wss 3723  𝒫 cpw 4297  cop 4322  {copab 4846  cmpt 4863  wf 6027  cfv 6031  2nd c2nd 7314  Vtxcvtx 26095  Edgcedg 26160  UPGraphcupgr 26196  USPGraphcuspgr 26265  Pairscspr 42255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-cda 9192  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-hash 13322  df-edg 26161  df-upgr 26198  df-uspgr 26267  df-spr 42256
This theorem is referenced by:  uspgrsprf1  42283  uspgrsprfo  42284
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