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Theorem clwlkfclwwlk1hashn 26368

Description: The size of the first component of a closed walk. (Contributed by Alexander van der Vekens, 5-Jul-2018.)

**Hypotheses**
Ref	Expression
clwlkfclwwlk.1	⊢ 𝐴 = (1^st ‘𝑐)
clwlkfclwwlk.2	⊢ 𝐵 = (2^nd ‘𝑐)
clwlkfclwwlk.c	⊢ 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
clwlkfclwwlk.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))

**Assertion**
Ref	Expression
clwlkfclwwlk1hashn	⊢ (𝑊 ∈ 𝐶 → (#‘(1^st ‘𝑊)) = 𝑁)

Distinct variable groups: 𝐸,𝑐 𝑁,𝑐 𝑉,𝑐 𝑊,𝑐 Allowed substitution hints: 𝐴(𝑐) 𝐵(𝑐) 𝐶(𝑐) 𝐹(𝑐)

**Proof of Theorem clwlkfclwwlk1hashn**
Step	Hyp	Ref	Expression
1		clwlkfclwwlk.1	. . . . . 6 ⊢ 𝐴 = (1^st ‘𝑐)
2	1	fveq2i 6106	. . . . 5 ⊢ (#‘𝐴) = (#‘(1^st ‘𝑐))
3	2	eqeq1i 2615	. . . 4 ⊢ ((#‘𝐴) = 𝑁 ↔ (#‘(1^st ‘𝑐)) = 𝑁)
4		fveq2 6103	. . . . . 6 ⊢ (𝑐 = 𝑊 → (1^st ‘𝑐) = (1^st ‘𝑊))
5	4	fveq2d 6107	. . . . 5 ⊢ (𝑐 = 𝑊 → (#‘(1^st ‘𝑐)) = (#‘(1^st ‘𝑊)))
6	5	eqeq1d 2612	. . . 4 ⊢ (𝑐 = 𝑊 → ((#‘(1^st ‘𝑐)) = 𝑁 ↔ (#‘(1^st ‘𝑊)) = 𝑁))
7	3, 6	syl5bb 271	. . 3 ⊢ (𝑐 = 𝑊 → ((#‘𝐴) = 𝑁 ↔ (#‘(1^st ‘𝑊)) = 𝑁))
8		clwlkfclwwlk.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}
9	7, 8	elrab2 3333	. 2 ⊢ (𝑊 ∈ 𝐶 ↔ (𝑊 ∈ (𝑉 ClWalks 𝐸) ∧ (#‘(1^st ‘𝑊)) = 𝑁))
10	9	simprbi 479	1 ⊢ (𝑊 ∈ 𝐶 → (#‘(1^st ‘𝑊)) = 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {crab 2900 ⟨cop 4131 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 0cc0 9815 #chash 12979 substr csubstr 13150 ClWalks cclwlk 26275

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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590

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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 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-iota 5768 df-fv 5812

This theorem is referenced by: clwlkf1clwwlklem 26376 clwlkf1clwwlk 26377

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