Theorem wlkwwlkfunOLD 27203

Description: Obsolete as of 2-Aug-2022. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 16-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
wlkwwlkbijOLD.t	⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}
wlkwwlkbijOLD.w	⊢ 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
wlkwwlkbijOLD.f	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))

Assertion

Ref	Expression
wlkwwlkfunOLD	⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)

Distinct variable groups:   𝐺,𝑝,𝑡,𝑤   𝑁,𝑝,𝑡,𝑤   𝑃,𝑝,𝑡,𝑤   𝑡,𝑇,𝑤   𝑡,𝑉   𝑡,𝑊

Allowed substitution hints:   𝑇(𝑝)   𝐹(𝑤,𝑡,𝑝)   𝑉(𝑤,𝑝)   𝑊(𝑤,𝑝)

Proof of Theorem wlkwwlkfunOLD

Step	Hyp	Ref	Expression
1		fveq2 6437	. . . . . . . 8 ⊢ (𝑝 = 𝑡 → (1st ‘𝑝) = (1st ‘𝑡))
2	1	fveq2d 6441	. . . . . . 7 ⊢ (𝑝 = 𝑡 → (♯‘(1st ‘𝑝)) = (♯‘(1st ‘𝑡)))
3	2	eqeq1d 2827	. . . . . 6 ⊢ (𝑝 = 𝑡 → ((♯‘(1st ‘𝑝)) = 𝑁 ↔ (♯‘(1st ‘𝑡)) = 𝑁))
4		fveq2 6437	. . . . . . . 8 ⊢ (𝑝 = 𝑡 → (2nd ‘𝑝) = (2nd ‘𝑡))
5	4	fveq1d 6439	. . . . . . 7 ⊢ (𝑝 = 𝑡 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑡)‘0))
6	5	eqeq1d 2827	. . . . . 6 ⊢ (𝑝 = 𝑡 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑡)‘0) = 𝑃))
7	3, 6	anbi12d 624	. . . . 5 ⊢ (𝑝 = 𝑡 → (((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)))
8		wlkwwlkbijOLD.t	. . . . 5 ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}
9	7, 8	elrab2 3589	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↔ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)))
10		simp1 1170	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐺 ∈ UPGraph)
11		simpl 476	. . . . . . 7 ⊢ ((𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)) → 𝑡 ∈ (Walks‘𝐺))
12	10, 11	anim12i 606	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → (𝐺 ∈ UPGraph ∧ 𝑡 ∈ (Walks‘𝐺)))
13		simp3 1172	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
14		simprl 787	. . . . . . 7 ⊢ ((𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)) → (♯‘(1st ‘𝑡)) = 𝑁)
15	13, 14	anim12i 606	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → (𝑁 ∈ ℕ0 ∧ (♯‘(1st ‘𝑡)) = 𝑁))
16		wlknewwlksn 27194	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑡 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st ‘𝑡)) = 𝑁)) → (2nd ‘𝑡) ∈ (𝑁 WWalksN 𝐺))
17	12, 15, 16	syl2anc 579	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → (2nd ‘𝑡) ∈ (𝑁 WWalksN 𝐺))
18		simprrr 800	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → ((2nd ‘𝑡)‘0) = 𝑃)
19	17, 18	jca 507	. . . 4 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → ((2nd ‘𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd ‘𝑡)‘0) = 𝑃))
20	9, 19	sylan2b 587	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑡 ∈ 𝑇) → ((2nd ‘𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd ‘𝑡)‘0) = 𝑃))
21		fveq1 6436	. . . . 5 ⊢ (𝑤 = (2nd ‘𝑡) → (𝑤‘0) = ((2nd ‘𝑡)‘0))
22	21	eqeq1d 2827	. . . 4 ⊢ (𝑤 = (2nd ‘𝑡) → ((𝑤‘0) = 𝑃 ↔ ((2nd ‘𝑡)‘0) = 𝑃))
23		wlkwwlkbijOLD.w	. . . 4 ⊢ 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
24	22, 23	elrab2 3589	. . 3 ⊢ ((2nd ‘𝑡) ∈ 𝑊 ↔ ((2nd ‘𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd ‘𝑡)‘0) = 𝑃))
25	20, 24	sylibr 226	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑡 ∈ 𝑇) → (2nd ‘𝑡) ∈ 𝑊)
26		wlkwwlkbijOLD.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))
27	25, 26	fmptd 6638	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  {crab 3121   ↦ cmpt 4954  ⟶wf 6123  ‘cfv 6127  (class class class)co 6910  1^st c1st 7431  2^nd c2nd 7432  0cc0 10259  ℕ₀cn0 11625  ♯chash 13417  UPGraphcupgr 26385  Walkscwlks 26901   WWalksN cwwlksn 27132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ifp 1090  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-om 7332  df-1st 7433  df-2nd 7434  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-1o 7831  df-2o 7832  df-oadd 7835  df-er 8014  df-map 8129  df-pm 8130  df-en 8229  df-dom 8230  df-sdom 8231  df-fin 8232  df-card 9085  df-cda 9312  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-nn 11358  df-2 11421  df-n0 11626  df-xnn0 11698  df-z 11712  df-uz 11976  df-fz 12627  df-fzo 12768  df-hash 13418  df-word 13582  df-edg 26353  df-uhgr 26363  df-upgr 26387  df-wlks 26904  df-wwlks 27136  df-wwlksn 27137

This theorem is referenced by:  wlkwwlkinjOLD  27204  wlkwwlksurOLD  27205