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Theorem pthon 26105
 Description: The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
Assertion
Ref Expression
pthon (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 PathOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})
Distinct variable groups:   𝑓,𝐸,𝑝   𝑓,𝑉,𝑝   𝑓,𝑋,𝑝   𝑓,𝑌,𝑝   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝

Proof of Theorem pthon
Dummy variables 𝑎 𝑏 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . . . . 5 (𝑉𝑋𝑉 ∈ V)
21ad2antrr 758 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → 𝑉 ∈ V)
3 elex 3185 . . . . . 6 (𝐸𝑌𝐸 ∈ V)
43adantl 481 . . . . 5 ((𝑉𝑋𝐸𝑌) → 𝐸 ∈ V)
54adantr 480 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → 𝐸 ∈ V)
6 id 22 . . . . . . 7 (𝑉𝑋𝑉𝑋)
76ancli 572 . . . . . 6 (𝑉𝑋 → (𝑉𝑋𝑉𝑋))
87ad2antrr 758 . . . . 5 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝑉𝑋𝑉𝑋))
9 mpt2exga 7135 . . . . 5 ((𝑉𝑋𝑉𝑋) → (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)}) ∈ V)
108, 9syl 17 . . . 4 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)}) ∈ V)
11 simpl 472 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
12 oveq12 6558 . . . . . . . . . 10 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 WalkOn 𝑒) = (𝑉 WalkOn 𝐸))
1312oveqd 6566 . . . . . . . . 9 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑎(𝑣 WalkOn 𝑒)𝑏) = (𝑎(𝑉 WalkOn 𝐸)𝑏))
1413breqd 4594 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝))
15 oveq12 6558 . . . . . . . . 9 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑣 Paths 𝑒) = (𝑉 Paths 𝐸))
1615breqd 4594 . . . . . . . 8 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑓(𝑣 Paths 𝑒)𝑝𝑓(𝑉 Paths 𝐸)𝑝))
1714, 16anbi12d 743 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Paths 𝑒)𝑝) ↔ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)))
1817opabbidv 4648 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Paths 𝑒)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})
1911, 11, 18mpt2eq123dv 6615 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Paths 𝑒)𝑝)}) = (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)}))
20 df-pthon 26044 . . . . 5 PathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Paths 𝑒)𝑝)}))
2119, 20ovmpt2ga 6688 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)}) ∈ V) → (𝑉 PathOn 𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)}))
222, 5, 10, 21syl3anc 1318 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝑉 PathOn 𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)}))
2322oveqd 6566 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 PathOn 𝐸)𝐵) = (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})𝐵))
24 simpl 472 . . . 4 ((𝐴𝑉𝐵𝑉) → 𝐴𝑉)
2524adantl 481 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → 𝐴𝑉)
26 simprr 792 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → 𝐵𝑉)
27 ancom 465 . . . . . . 7 ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝) ↔ (𝑓(𝑉 Paths 𝐸)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝))
2827a1i 11 . . . . . 6 ((𝑉𝑋𝐸𝑌) → ((𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝) ↔ (𝑓(𝑉 Paths 𝐸)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)))
2928opabbidv 4648 . . . . 5 ((𝑉𝑋𝐸𝑌) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Paths 𝐸)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)})
30 pthistrl 26102 . . . . . . . 8 (𝑓(𝑉 Paths 𝐸)𝑝𝑓(𝑉 Trails 𝐸)𝑝)
31 trliswlk 26069 . . . . . . . 8 (𝑓(𝑉 Trails 𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)
3230, 31syl 17 . . . . . . 7 (𝑓(𝑉 Paths 𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)
3332wlkres 26050 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Paths 𝐸)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)} ∈ V)
341, 3, 33syl2an 493 . . . . 5 ((𝑉𝑋𝐸𝑌) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Paths 𝐸)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝)} ∈ V)
3529, 34eqeltrd 2688 . . . 4 ((𝑉𝑋𝐸𝑌) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)} ∈ V)
3635adantr 480 . . 3 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)} ∈ V)
37 oveq12 6558 . . . . . . 7 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑎(𝑉 WalkOn 𝐸)𝑏) = (𝐴(𝑉 WalkOn 𝐸)𝐵))
3837breqd 4594 . . . . . 6 ((𝑎 = 𝐴𝑏 = 𝐵) → (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝))
3938anbi1d 737 . . . . 5 ((𝑎 = 𝐴𝑏 = 𝐵) → ((𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝) ↔ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)))
4039opabbidv 4648 . . . 4 ((𝑎 = 𝐴𝑏 = 𝐵) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)} = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})
41 eqid 2610 . . . 4 (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)}) = (𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})
4240, 41ovmpt2ga 6688 . . 3 ((𝐴𝑉𝐵𝑉 ∧ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)} ∈ V) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})
4325, 26, 36, 42syl3anc 1318 . 2 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑎𝑉, 𝑏𝑉 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})
4423, 43eqtrd 2644 1 (((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 PathOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Paths 𝐸)𝑝)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  {copab 4642  (class class class)co 6549   ↦ cmpt2 6551   Walks cwalk 26026   Trails ctrail 26027   Paths cpath 26028   WalkOn cwlkon 26030   PathOn cpthon 26032 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036  df-trail 26037  df-pth 26038  df-pthon 26044 This theorem is referenced by:  ispthon  26106  pthonprop  26107
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