Theorem wlkwwlkbij2OLD 27200

Description: Obsolete version of wlksnwwlknvbij 27230 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.)

Assertion

Ref	Expression
wlkwwlkbij2OLD	⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})

Distinct variable groups:   𝐺,𝑝   𝑤,𝐺,𝑓   𝑓,𝑝,𝑁   𝑤,𝑁,𝑓   𝑃,𝑝   𝑤,𝑃,𝑓

Allowed substitution hints:   𝑉(𝑤,𝑓,𝑝)

Proof of Theorem wlkwwlkbij2OLD

Dummy variables 𝑠 𝑢 𝑥 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6446	. . 3 ⊢ (Walks‘𝐺) ∈ V
2	1	mptrabex 6744	. 2 ⊢ (𝑥 ∈ {𝑡 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)) ∈ V
3		fveq2 6433	. . . . . . 7 ⊢ (𝑝 = 𝑢 → (1st ‘𝑝) = (1st ‘𝑢))
4	3	fveq2d 6437	. . . . . 6 ⊢ (𝑝 = 𝑢 → (♯‘(1st ‘𝑝)) = (♯‘(1st ‘𝑢)))
5	4	eqeq1d 2827	. . . . 5 ⊢ (𝑝 = 𝑢 → ((♯‘(1st ‘𝑝)) = 𝑁 ↔ (♯‘(1st ‘𝑢)) = 𝑁))
6		fveq2 6433	. . . . . . 7 ⊢ (𝑝 = 𝑢 → (2nd ‘𝑝) = (2nd ‘𝑢))
7	6	fveq1d 6435	. . . . . 6 ⊢ (𝑝 = 𝑢 → ((2nd ‘𝑝)‘0) = ((2nd ‘𝑢)‘0))
8	7	eqeq1d 2827	. . . . 5 ⊢ (𝑝 = 𝑢 → (((2nd ‘𝑝)‘0) = 𝑃 ↔ ((2nd ‘𝑢)‘0) = 𝑃))
9	5, 8	anbi12d 626	. . . 4 ⊢ (𝑝 = 𝑢 → (((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃) ↔ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)))
10	9	cbvrabv 3412	. . 3 ⊢ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} = {𝑢 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑢)) = 𝑁 ∧ ((2nd ‘𝑢)‘0) = 𝑃)}
11		fveq1 6432	. . . . 5 ⊢ (𝑤 = 𝑠 → (𝑤‘0) = (𝑠‘0))
12	11	eqeq1d 2827	. . . 4 ⊢ (𝑤 = 𝑠 → ((𝑤‘0) = 𝑃 ↔ (𝑠‘0) = 𝑃))
13	12	cbvrabv 3412	. . 3 ⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} = {𝑠 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑠‘0) = 𝑃}
14		fveq2 6433	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → (1st ‘𝑡) = (1st ‘𝑝))
15	14	fveq2d 6437	. . . . . . 7 ⊢ (𝑡 = 𝑝 → (♯‘(1st ‘𝑡)) = (♯‘(1st ‘𝑝)))
16	15	eqeq1d 2827	. . . . . 6 ⊢ (𝑡 = 𝑝 → ((♯‘(1st ‘𝑡)) = 𝑁 ↔ (♯‘(1st ‘𝑝)) = 𝑁))
17		fveq2 6433	. . . . . . . 8 ⊢ (𝑡 = 𝑝 → (2nd ‘𝑡) = (2nd ‘𝑝))
18	17	fveq1d 6435	. . . . . . 7 ⊢ (𝑡 = 𝑝 → ((2nd ‘𝑡)‘0) = ((2nd ‘𝑝)‘0))
19	18	eqeq1d 2827	. . . . . 6 ⊢ (𝑡 = 𝑝 → (((2nd ‘𝑡)‘0) = 𝑃 ↔ ((2nd ‘𝑝)‘0) = 𝑃))
20	16, 19	anbi12d 626	. . . . 5 ⊢ (𝑡 = 𝑝 → (((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃) ↔ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)))
21	20	cbvrabv 3412	. . . 4 ⊢ {𝑡 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} = {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}
22	21	mpteq1i 4962	. . 3 ⊢ (𝑥 ∈ {𝑡 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)) = (𝑥 ∈ {𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)} ↦ (2nd ‘𝑥))
23	10, 13, 22	wlkwwlkbijOLD 27199	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑥 ∈ {𝑡 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)):{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})
24		f1oeq1 6367	. . 3 ⊢ (𝑓 = (𝑥 ∈ {𝑡 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)) → (𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} ↔ (𝑥 ∈ {𝑡 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)):{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
25	24	spcegv 3511	. 2 ⊢ ((𝑥 ∈ {𝑡 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)) ∈ V → ((𝑥 ∈ {𝑡 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑡)) = 𝑁 ∧ ((2nd ‘𝑡)‘0) = 𝑃)} ↦ (2nd ‘𝑥)):{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃} → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
26	2, 23, 25	mpsyl 68	1 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1113   = wceq 1658  ∃wex 1880   ∈ wcel 2166  {crab 3121  Vcvv 3414   ↦ cmpt 4952  –1-1-onto→wf1o 6122  ‘cfv 6123  (class class class)co 6905  1^st c1st 7426  2^nd c2nd 7427  0cc0 10252  ℕ₀cn0 11618  ♯chash 13410  USPGraphcuspgr 26447  Walkscwlks 26894   WWalksN cwwlksn 27125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  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 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-card 9078  df-cda 9305  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-n0 11619  df-xnn0 11691  df-z 11705  df-uz 11969  df-fz 12620  df-fzo 12761  df-hash 13411  df-word 13575  df-edg 26346  df-uhgr 26356  df-upgr 26380  df-uspgr 26449  df-wlks 26897  df-wwlks 27129  df-wwlksn 27130

This theorem is referenced by: (None)