rusgranumwlklem4 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > rusgranumwlklem4		Structured version Visualization version GIF version

Theorem rusgranumwlklem4 26479

Description: Lemma 4 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 24-Jul-2018.) (Proof shortened by AV, 5-May-2021.)

**Hypotheses**
Ref	Expression
rusgranumwlk.w	⊢ 𝑊 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑛})
rusgranumwlk.l	⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ₀ ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2^nd ‘𝑤)‘0) = 𝑣}))

**Assertion**
Ref	Expression
rusgranumwlklem4	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))

Distinct variable groups: 𝐸,𝑐,𝑛 𝑁,𝑐,𝑛 𝑉,𝑐,𝑛 𝑣,𝑁,𝑤 𝑃,𝑛,𝑣,𝑤 𝑣,𝑉 𝑛,𝑊,𝑣,𝑤 𝑤,𝑉,𝑐 𝑣,𝐸,𝑤 Allowed substitution hints: 𝑃(𝑐) 𝐿(𝑤,𝑣,𝑛,𝑐) 𝑊(𝑐)

Proof of Theorem rusgranumwlklem4 Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rusgranumwlk.w	. . . 4 ⊢ 𝑊 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑛})
2		rusgranumwlk.l	. . . 4 ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ₀ ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2^nd ‘𝑤)‘0) = 𝑣}))
3	1, 2	rusgranumwlklem3 26478	. . 3 ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑃)}))
4	3	3adant1 1072	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑃)}))
5		fveq2 6103	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (1^st ‘𝑤) = (1^st ‘𝑣))
6	5	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 = 𝑣 → (#‘(1^st ‘𝑤)) = (#‘(1^st ‘𝑣)))
7	6	eqeq1d 2612	. . . . . 6 ⊢ (𝑤 = 𝑣 → ((#‘(1^st ‘𝑤)) = 𝑁 ↔ (#‘(1^st ‘𝑣)) = 𝑁))
8		fveq2 6103	. . . . . . . 8 ⊢ (𝑤 = 𝑣 → (2^nd ‘𝑤) = (2^nd ‘𝑣))
9	8	fveq1d 6105	. . . . . . 7 ⊢ (𝑤 = 𝑣 → ((2^nd ‘𝑤)‘0) = ((2^nd ‘𝑣)‘0))
10	9	eqeq1d 2612	. . . . . 6 ⊢ (𝑤 = 𝑣 → (((2^nd ‘𝑤)‘0) = 𝑃 ↔ ((2^nd ‘𝑣)‘0) = 𝑃))
11	7, 10	anbi12d 743	. . . . 5 ⊢ (𝑤 = 𝑣 → (((#‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑃) ↔ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)))
12	11	cbvrabv 3172	. . . 4 ⊢ {𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑃)} = {𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)}
13	12	a1i 11	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → {𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑃)} = {𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)})
14	13	fveq2d 6107	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑃)}) = (#‘{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)}))
15		ovex 6577	. . . 4 ⊢ (𝑉 Walks 𝐸) ∈ V
16	15	rabex 4740	. . 3 ⊢ {𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)} ∈ V
17		wlkiswwlkbij2 26249	. . 3 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → ∃𝑓 𝑓:{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃})
18		hasheqf1oi 13002	. . 3 ⊢ ({𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)} ∈ V → (∃𝑓 𝑓:{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃} → (#‘{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)}) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃})))
19	16, 17, 18	mpsyl 66	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (#‘{𝑣 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1^st ‘𝑣)) = 𝑁 ∧ ((2^nd ‘𝑣)‘0) = 𝑃)}) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
20	4, 14, 19	3eqtrd 2648	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ₀) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {crab 2900 Vcvv 3173 class class class wbr 4583 ↦ cmpt 4643 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058 0cc0 9815 ℕ₀cn0 11169 #chash 12979 USGrph cusg 25859 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-fal 1481 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-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-usgra 25862 df-wlk 26036 df-wwlk 26207 df-wwlkn 26208

This theorem is referenced by: rusgranumwlkb0 26480 rusgranumwlkb1 26481 rusgra0edg 26482 rusgranumwlks 26483

W3C validator