Theorem wlkiswwlkfun 26245

Description: Lemma 1 for wlkiswwlkbij2 26249. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem wlkiswwlkfun

Step	Hyp	Ref	Expression
1		fveq2 6103	. . . . . . . 8 ⊢ (𝑝 = 𝑡 → (1st ‘𝑝) = (1st ‘𝑡))
2	1	fveq2d 6107	. . . . . . 7 ⊢ (𝑝 = 𝑡 → (#‘(1st ‘𝑝)) = (#‘(1st ‘𝑡)))
3	2	eqeq1d 2612	. . . . . 6 ⊢ (𝑝 = 𝑡 → ((#‘(1st ‘𝑝)) = 𝑁 ↔ (#‘(1st ‘𝑡)) = 𝑁))
4		fveq2 6103	. . . . . . . 8 ⊢ (𝑝 = 𝑡 → (2nd ‘𝑝) = (2nd ‘𝑡))
5	4	fveq1d 6105	. . . . . . 7 ⊢ (𝑝 = 𝑡 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑡)‘0))
6	5	eqeq1d 2612	. . . . . 6 ⊢ (𝑝 = 𝑡 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑡)‘0) = 𝑃))
7	3, 6	anbi12d 743	. . . . 5 ⊢ (𝑝 = 𝑡 → (((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)))
8		wlkiswwlkbij.t	. . . . 5 ⊢ 𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}
9	7, 8	elrab2 3333	. . . 4 ⊢ (𝑡 ∈ 𝑇 ↔ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)))
10		simpr 476	. . . . . 6 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
11		simpl 472	. . . . . . 7 ⊢ (((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃) → (#‘(1st ‘𝑡)) = 𝑁)
12	11	anim2i 591	. . . . . 6 ⊢ ((𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)) → (𝑡 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑡)) = 𝑁))
13		vfwlkniswwlkn 26234	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑡)) = 𝑁)) → (2nd ‘𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
14	10, 12, 13	syl2an 493	. . . . 5 ⊢ (((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → (2nd ‘𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))
15		simprrr 801	. . . . 5 ⊢ (((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → ((2nd ‘𝑡)‘0) = 𝑃)
16	14, 15	jca 553	. . . 4 ⊢ (((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ (𝑡 ∈ (𝑉 Walks 𝐸) ∧ ((#‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃))) → ((2nd ‘𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((2nd ‘𝑡)‘0) = 𝑃))
17	9, 16	sylan2b 491	. . 3 ⊢ (((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑡 ∈ 𝑇) → ((2nd ‘𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((2nd ‘𝑡)‘0) = 𝑃))
18		fveq1 6102	. . . . 5 ⊢ (𝑤 = (2nd ‘𝑡) → (𝑤‘0) = ((2nd ‘𝑡)‘0))
19	18	eqeq1d 2612	. . . 4 ⊢ (𝑤 = (2nd ‘𝑡) → ((𝑤‘0) = 𝑃 ↔ ((2nd ‘𝑡)‘0) = 𝑃))
20		wlkiswwlkbij.w	. . . 4 ⊢ 𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
21	19, 20	elrab2 3333	. . 3 ⊢ ((2nd ‘𝑡) ∈ 𝑊 ↔ ((2nd ‘𝑡) ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ ((2nd ‘𝑡)‘0) = 𝑃))
22	17, 21	sylibr 223	. 2 ⊢ (((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) ∧ 𝑡 ∈ 𝑇) → (2nd ‘𝑡) ∈ 𝑊)
23		wlkiswwlkbij.f	. 2 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))
24	22, 23	fmptd 6292	1 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇⟶𝑊)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058  0cc0 9815  ℕ₀cn0 11169  #chash 12979   Walks cwalk 26026   WWalksN cwwlkn 26206

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-wwlk 26207  df-wwlkn 26208

This theorem is referenced by:  wlkiswwlkinj  26246  wlkiswwlksur  26247