Theorem uspgrsprf 48138

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 2821	. . . 4 ⊢ (𝑔 ∈ 𝐺 ↔ 𝑔 ∈ {⟨𝑣, 𝑒⟩ ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))})
4		elopab 5490	. . . 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 275	. . 3 ⊢ (𝑔 ∈ 𝐺 ↔ ∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))))
6		uspgrupgr 29112	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ USPGraph → 𝑞 ∈ UPGraph)
7		upgredgssspr 48135	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ UPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
8	6, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ USPGraph → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
9	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → (Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)))
10		simpr 484	. . . . . . . . . . . . 13 ⊢ (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → (Edg‘𝑞) = 𝑒)
11		fveq2 6861	. . . . . . . . . . . . . 14 ⊢ ((Vtx‘𝑞) = 𝑣 → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
12	11	adantr 480	. . . . . . . . . . . . 13 ⊢ (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → (Pairs‘(Vtx‘𝑞)) = (Pairs‘𝑣))
13	10, 12	sseq12d 3983	. . . . . . . . . . . 12 ⊢ (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → ((Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)) ↔ 𝑒 ⊆ (Pairs‘𝑣)))
14	13	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → ((Edg‘𝑞) ⊆ (Pairs‘(Vtx‘𝑞)) ↔ 𝑒 ⊆ (Pairs‘𝑣)))
15	9, 14	mpbid 232	. . . . . . . . . 10 ⊢ ((𝑞 ∈ USPGraph ∧ ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑣))
16	15	rexlimiva 3127	. . . . . . . . 9 ⊢ (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) → 𝑒 ⊆ (Pairs‘𝑣))
17	16	adantl 481	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑣))
18		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑣 = 𝑉 → (Pairs‘𝑣) = (Pairs‘𝑉))
19	18	sseq2d 3982	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (𝑒 ⊆ (Pairs‘𝑣) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → (𝑒 ⊆ (Pairs‘𝑣) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
21	17, 20	mpbid 232	. . . . . . 7 ⊢ ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) → 𝑒 ⊆ (Pairs‘𝑉))
22	21	adantl 481	. . . . . 6 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → 𝑒 ⊆ (Pairs‘𝑉))
23		vex 3454	. . . . . . . . 9 ⊢ 𝑣 ∈ V
24		vex 3454	. . . . . . . . 9 ⊢ 𝑒 ∈ V
25	23, 24	op2ndd 7982	. . . . . . . 8 ⊢ (𝑔 = ⟨𝑣, 𝑒⟩ → (2nd ‘𝑔) = 𝑒)
26	25	sseq1d 3981	. . . . . . 7 ⊢ (𝑔 = ⟨𝑣, 𝑒⟩ → ((2nd ‘𝑔) ⊆ (Pairs‘𝑉) ↔ 𝑒 ⊆ (Pairs‘𝑉)))
27	26	adantr 480	. . . . . 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 257	. . . . 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 2821	. . . . . 6 ⊢ ((2nd ‘𝑔) ∈ 𝑃 ↔ (2nd ‘𝑔) ∈ 𝒫 (Pairs‘𝑉))
31		fvex 6874	. . . . . . 7 ⊢ (2nd ‘𝑔) ∈ V
32	31	elpw 4570	. . . . . 6 ⊢ ((2nd ‘𝑔) ∈ 𝒫 (Pairs‘𝑉) ↔ (2nd ‘𝑔) ⊆ (Pairs‘𝑉))
33	30, 32	bitri 275	. . . . 5 ⊢ ((2nd ‘𝑔) ∈ 𝑃 ↔ (2nd ‘𝑔) ⊆ (Pairs‘𝑉))
34	28, 33	sylibr 234	. . . 4 ⊢ ((𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd ‘𝑔) ∈ 𝑃)
35	34	exlimivv 1932	. . 3 ⊢ (∃𝑣∃𝑒(𝑔 = ⟨𝑣, 𝑒⟩ ∧ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))) → (2nd ‘𝑔) ∈ 𝑃)
36	5, 35	sylbi 217	. 2 ⊢ (𝑔 ∈ 𝐺 → (2nd ‘𝑔) ∈ 𝑃)
37	1, 36	fmpti 7087	1 ⊢ 𝐹:𝐺⟶𝑃

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054   ⊆ wss 3917  𝒫 cpw 4566  ⟨cop 4598  {copab 5172   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  2^nd c2nd 7970  Vtxcvtx 28930  Edgcedg 28981  UPGraphcupgr 29014  USPGraphcuspgr 29082  Pairscspr 47482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-oadd 8441  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-n0 12450  df-xnn0 12523  df-z 12537  df-uz 12801  df-fz 13476  df-hash 14303  df-edg 28982  df-upgr 29016  df-uspgr 29084  df-spr 47483

This theorem is referenced by:  uspgrsprf1  48139  uspgrsprfo  48140