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Theorem rusgranumwlklem1 26476

Description: Lemma 1 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 21-Jul-2018.)

**Hypothesis**
Ref	Expression
rusgranumwlk.w	⊢ 𝑊 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑛})

**Assertion**
Ref	Expression
rusgranumwlklem1	⊢ (𝑅 ∈ (𝑊‘𝑁) → (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1^st ‘𝑅)) = 𝑁))

Distinct variable groups: 𝐸,𝑐,𝑛 𝑁,𝑐,𝑛 𝑅,𝑐 𝑉,𝑐,𝑛 Allowed substitution hints: 𝑅(𝑛) 𝑊(𝑛,𝑐)

**Proof of Theorem rusgranumwlklem1**
Step	Hyp	Ref	Expression
1		rusgranumwlk.w	. . 3 ⊢ 𝑊 = (𝑛 ∈ ℕ₀ ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑛})
2		ovex 6577	. . . 4 ⊢ (𝑉 Walks 𝐸) ∈ V
3	2	a1i 11	. . 3 ⊢ (𝑁 ∈ ℕ₀ → (𝑉 Walks 𝐸) ∈ V)
4	1, 3	elfvmptrab 6214	. 2 ⊢ (𝑅 ∈ (𝑊‘𝑁) → (𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ (𝑉 Walks 𝐸)))
5	2	rabex 4740	. . . . . . 7 ⊢ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑁} ∈ V
6	5	a1i 11	. . . . . 6 ⊢ (𝑅 ∈ (𝑉 Walks 𝐸) → {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑁} ∈ V)
7		eqeq2 2621	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((#‘(1^st ‘𝑐)) = 𝑛 ↔ (#‘(1^st ‘𝑐)) = 𝑁))
8	7	rabbidv 3164	. . . . . . 7 ⊢ (𝑛 = 𝑁 → {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑛} = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑁})
9	8, 1	fvmptg 6189	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑁} ∈ V) → (𝑊‘𝑁) = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑁})
10	6, 9	sylan2 490	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑊‘𝑁) = {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑁})
11	10	eleq2d 2673	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑅 ∈ (𝑊‘𝑁) ↔ 𝑅 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑁}))
12		fveq2 6103	. . . . . . . 8 ⊢ (𝑐 = 𝑅 → (1^st ‘𝑐) = (1^st ‘𝑅))
13	12	fveq2d 6107	. . . . . . 7 ⊢ (𝑐 = 𝑅 → (#‘(1^st ‘𝑐)) = (#‘(1^st ‘𝑅)))
14	13	eqeq1d 2612	. . . . . 6 ⊢ (𝑐 = 𝑅 → ((#‘(1^st ‘𝑐)) = 𝑁 ↔ (#‘(1^st ‘𝑅)) = 𝑁))
15	14	elrab 3331	. . . . 5 ⊢ (𝑅 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑁} ↔ (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1^st ‘𝑅)) = 𝑁))
16		simpr 476	. . . . . 6 ⊢ ((𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1^st ‘𝑅)) = 𝑁) → (#‘(1^st ‘𝑅)) = 𝑁)
17	16	a1i 11	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ (𝑉 Walks 𝐸)) → ((𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1^st ‘𝑅)) = 𝑁) → (#‘(1^st ‘𝑅)) = 𝑁))
18	15, 17	syl5bi 231	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑅 ∈ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1^st ‘𝑐)) = 𝑁} → (#‘(1^st ‘𝑅)) = 𝑁))
19	11, 18	sylbid 229	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑅 ∈ (𝑊‘𝑁) → (#‘(1^st ‘𝑅)) = 𝑁))
20		simpr 476	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ (𝑉 Walks 𝐸)) → 𝑅 ∈ (𝑉 Walks 𝐸))
21	19, 20	jctild 564	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑅 ∈ (𝑉 Walks 𝐸)) → (𝑅 ∈ (𝑊‘𝑁) → (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1^st ‘𝑅)) = 𝑁)))
22	4, 21	mpcom 37	1 ⊢ (𝑅 ∈ (𝑊‘𝑁) → (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1^st ‘𝑅)) = 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 ℕ₀cn0 11169 #chash 12979 Walks cwalk 26026

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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 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-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552

This theorem is referenced by: (None)

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