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Theorem clwlksfclwwlk1hashn 26959

Description: The size of the first component of a closed walk. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 2-May-2021.)

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

**Assertion**
Ref	Expression
clwlksfclwwlk1hashn	⊢ (𝑊 ∈ 𝐶 → (#‘(1^st ‘𝑊)) = 𝑁)

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

**Proof of Theorem clwlksfclwwlk1hashn**
Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . . . . 6 ⊢ 𝐴 = (1^st ‘𝑐)
2	1	fveq2i 6194	. . . . 5 ⊢ (#‘𝐴) = (#‘(1^st ‘𝑐))
3	2	eqeq1i 2627	. . . 4 ⊢ ((#‘𝐴) = 𝑁 ↔ (#‘(1^st ‘𝑐)) = 𝑁)
4		fveq2 6191	. . . . . 6 ⊢ (𝑐 = 𝑊 → (1^st ‘𝑐) = (1^st ‘𝑊))
5	4	fveq2d 6195	. . . . 5 ⊢ (𝑐 = 𝑊 → (#‘(1^st ‘𝑐)) = (#‘(1^st ‘𝑊)))
6	5	eqeq1d 2624	. . . 4 ⊢ (𝑐 = 𝑊 → ((#‘(1^st ‘𝑐)) = 𝑁 ↔ (#‘(1^st ‘𝑊)) = 𝑁))
7	3, 6	syl5bb 272	. . 3 ⊢ (𝑐 = 𝑊 → ((#‘𝐴) = 𝑁 ↔ (#‘(1^st ‘𝑊)) = 𝑁))
8		clwlksfclwwlk.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}
9	7, 8	elrab2 3366	. 2 ⊢ (𝑊 ∈ 𝐶 ↔ (𝑊 ∈ (ClWalks‘𝐺) ∧ (#‘(1^st ‘𝑊)) = 𝑁))
10	9	simprbi 480	1 ⊢ (𝑊 ∈ 𝐶 → (#‘(1^st ‘𝑊)) = 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 {crab 2916 ⟨cop 4183 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 0cc0 9936 #chash 13117 substr csubstr 13295 ClWalkscclwlks 26666

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896

This theorem is referenced by: clwlksf1clwwlklem 26968 clwlksf1clwwlk 26969

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