Theorem wlkwwlkfun 26781

Description: Lemma 1 for wlkwwlkbij2 26785. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)

Hypotheses

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

Assertion

Ref	Expression
wlkwwlkfun	⊢ ((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 wlkwwlkfun

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . . 8 ⊢ (𝑝 = 𝑡 → (1st ‘𝑝) = (1st ‘𝑡))
2	1	fveq2d 6195	. . . . . . 7 ⊢ (𝑝 = 𝑡 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑡)))
3	2	eqeq1d 2624	. . . . . 6 ⊢ (𝑝 = 𝑡 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑡)) = 𝑁))
4		fveq2 6191	. . . . . . . 8 ⊢ (𝑝 = 𝑡 → (2nd ‘𝑝) = (2nd ‘𝑡))
5	4	fveq1d 6193	. . . . . . 7 ⊢ (𝑝 = 𝑡 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑡)‘0))
6	5	eqeq1d 2624	. . . . . 6 ⊢ (𝑝 = 𝑡 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑡)‘0) = 𝑃))
7	3, 6	anbi12d 747	. . . . 5 ⊢ (𝑝 = 𝑡 → (((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)))
8		wlkwwlkbij.t	. . . . 5 ⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}
9	7, 8	elrab2 3366	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↔ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)))
10		simp1 1061	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐺 ∈ UPGraph )
11		simpl 473	. . . . . . 7 ⊢ ((𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)) → 𝑡 ∈ (Walks‘𝐺))
12	10, 11	anim12i 590	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → (𝐺 ∈ UPGraph ∧ 𝑡 ∈ (Walks‘𝐺)))
13		simp3 1063	. . . . . . 7 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
14		simprl 794	. . . . . . 7 ⊢ ((𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)) → (#‘(1st ‘𝑡)) = 𝑁)
15	13, 14	anim12i 590	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → (𝑁 ∈ ℕ0 ∧ (#‘(1st ‘𝑡)) = 𝑁))
16		wlknewwlksn 26773	. . . . . 6 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑡 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (#‘(1st ‘𝑡)) = 𝑁)) → (2nd ‘𝑡) ∈ (𝑁 WWalksN 𝐺))
17	12, 15, 16	syl2anc 693	. . . . 5 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (Walks‘𝐺) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → (2nd ‘𝑡) ∈ (𝑁 WWalksN 𝐺))
18		simprrr 805	. . . . 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 554	. . . 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 492	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑡 ∈ 𝑇) → ((2nd ‘𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd ‘𝑡)‘0) = 𝑃))
21		fveq1 6190	. . . . 5 ⊢ (𝑤 = (2nd ‘𝑡) → (𝑤‘0) = ((2nd ‘𝑡)‘0))
22	21	eqeq1d 2624	. . . 4 ⊢ (𝑤 = (2nd ‘𝑡) → ((𝑤‘0) = 𝑃 ↔ ((2nd ‘𝑡)‘0) = 𝑃))
23		wlkwwlkbij.w	. . . 4 ⊢ 𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
24	22, 23	elrab2 3366	. . 3 ⊢ ((2nd ‘𝑡) ∈ 𝑊 ↔ ((2nd ‘𝑡) ∈ (𝑁 WWalksN 𝐺) ∧ ((2nd ‘𝑡)‘0) = 𝑃))
25	20, 24	sylibr 224	. 2 ⊢ (((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑡 ∈ 𝑇) → (2nd ‘𝑡) ∈ 𝑊)
26		wlkwwlkbij.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))
27	25, 26	fmptd 6385	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  0cc0 9936  ℕ₀cn0 11292  #chash 13117   UPGraph cupgr 25975  Walkscwlks 26492   WWalksN cwwlksn 26718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-edg 25940  df-uhgr 25953  df-upgr 25977  df-wlks 26495  df-wwlks 26722  df-wwlksn 26723

This theorem is referenced by:  wlkwwlkinj  26782  wlkwwlksur  26783