Theorem uspgrsprf 48637

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 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))

Assertion

Ref	Expression
uspgrsprf	⊢ 𝐹:𝐺⟶𝑃

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

Step	Hyp	Ref	Expression
1		uspgrsprf.f	. 2 ⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔))
2		uspgrsprf.g	. . . . 5 ⊢ 𝐺 = {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}
3	2	eleq2i 2831	. . . 4 ⊢ (𝑔 ∈ 𝐺 ↔ 𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
4		elopab 5469	. . . 4 ⊢ (𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ ∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
5	3, 4	bitri 276	. . 3 ⊢ (𝑔 ∈ 𝐺 ↔ ∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
6		uspgrupgr 29265	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ USPGraph → 𝑞 ∈ UPGraph)
7		upgredgssspr 48634	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ UPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
8	6, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ USPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
9	8	adantr 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 6827	. . . . . . . . . . . . . 14 ⊢ ((Vtx‘𝑞) = 𝑣 → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
12	11	adantr 481	. . . . . . . . . . . . 13 ⊢ (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
13	10, 12	sseq12d 3948	. . . . . . . . . . . 12 ⊢ (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → ((Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)) ↔ 𝑒 ⊆ (Pairs‘𝑣)))
14	13	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → ((Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)) ↔ 𝑒 ⊆ (Pairs‘𝑣)))
15	9, 14	mpbid 233	. . . . . . . . . 10 ⊢ ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑣))
16	15	rexlimiva 3132	. . . . . . . . 9 ⊢ (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → 𝑒 ⊆ (Pairs‘𝑣))
17	16	adantl 482	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑣))
18		fveq2 6827	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → (Pairs‘𝑣) = (Pairs‘𝑉))
19	18	sseq2d 3947	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (𝑒 ⊆ (Pairs‘𝑣) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
20	19	adantr 481	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → (𝑒 ⊆ (Pairs‘𝑣) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
21	17, 20	mpbid 233	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑉))
22	21	adantl 482	. . . . . 6 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → 𝑒 ⊆ (Pairs‘𝑉))
23		vex 3435	. . . . . . . . 9 ⊢ 𝑣 ∈ V
24		vex 3435	. . . . . . . . 9 ⊢ 𝑒 ∈ V
25	23, 24	op2ndd 7942	. . . . . . . 8 ⊢ (𝑔 = ⟨𝑣, 𝑒⟩ → (2nd ‘𝑔) = 𝑒)
26	25	sseq1d 3946	. . . . . . 7 ⊢ (𝑔 = ⟨𝑣, 𝑒⟩ → ((2nd ‘𝑔) ⊆ (Pairs‘𝑉) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
27	26	adantr 481	. . . . . 6 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → ((2nd ‘𝑔) ⊆ (Pairs‘𝑉) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
28	22, 27	mpbird 258	. . . . 5 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd ‘𝑔) ⊆ (Pairs‘𝑉))
29		uspgrsprf.p	. . . . . . 7 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
30	29	eleq2i 2831	. . . . . 6 ⊢ ((2nd ‘𝑔) ∈ 𝑃 ↔ (2nd ‘𝑔) ∈ 𝒫 (Pairs‘𝑉))
31		fvex 6840	. . . . . . 7 ⊢ (2nd ‘𝑔) ∈ V
32	31	elpw 4533	. . . . . 6 ⊢ ((2nd ‘𝑔) ∈ 𝒫 (Pairs‘𝑉) ↔ (2nd ‘𝑔) ⊆ (Pairs‘𝑉))
33	30, 32	bitri 276	. . . . 5 ⊢ ((2nd ‘𝑔) ∈ 𝑃 ↔ (2nd ‘𝑔) ⊆ (Pairs‘𝑉))
34	28, 33	sylibr 235	. . . 4 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd ‘𝑔) ∈ 𝑃)
35	34	exlimivv 1939	. . 3 ⊢ (∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd ‘𝑔) ∈ 𝑃)
36	5, 35	sylbi 218	. 2 ⊢ (𝑔 ∈ 𝐺 → (2nd ‘𝑔) ∈ 𝑃)
37	1, 36	fmpti 7053	1 ⊢ 𝐹:𝐺⟶𝑃

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∃wrex 3063   ⊆ wss 3883  𝒫 cpw 4529  ⟨cop 4561  {copab 5134   ↦ cmpt 5153  ⟶wf 6481  ‘cfv 6485  2^nd c2nd 7930  Vtxcvtx 29083  Edgcedg 29134  UPGraphcupgr 29167  USPGraphcuspgr 29235  Pairscspr 47952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9816  df-card 9854  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-xnn0 12502  df-z 12516  df-uz 12780  df-fz 13453  df-hash 14284  df-edg 29135  df-upgr 29169  df-uspgr 29237  df-spr 47953

This theorem is referenced by:  uspgrsprf1  48638  uspgrsprfo  48639