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Theorem 2spotfi 26419

Description: In a finite graph, the set of simple paths of length 2 between two vertices (as ordered triples) is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.)

**Assertion**
Ref	Expression
2spotfi	⊢ (((𝑉 ∈ Fin ∧ 𝐸 ∈ 𝑋) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) ∈ Fin)

Proof of Theorem 2spotfi Dummy variables 𝑓 𝑝 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2spthonot 26393	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐸 ∈ 𝑋) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐵))})
2		3xpfi 8117	. . . 4 ⊢ (𝑉 ∈ Fin → ((𝑉 × 𝑉) × 𝑉) ∈ Fin)
3		rabfi 8070	. . . 4 ⊢ (((𝑉 × 𝑉) × 𝑉) ∈ Fin → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐵))} ∈ Fin)
4	2, 3	syl 17	. . 3 ⊢ (𝑉 ∈ Fin → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐵))} ∈ Fin)
5	4	ad2antrr 758	. 2 ⊢ (((𝑉 ∈ Fin ∧ 𝐸 ∈ 𝑋) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓∃𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐵))} ∈ Fin)
6	1, 5	eqeltrd 2688	1 ⊢ (((𝑉 ∈ Fin ∧ 𝐸 ∈ 𝑋) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {crab 2900 class class class wbr 4583 × cxp 5036 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 Fincfn 7841 1c1 9816 2c2 10947 #chash 12979 SPathOn cspthon 26033 2SPathOnOt c2pthonot 26384

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

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-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-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-en 7842 df-fin 7845 df-2spthonot 26387

This theorem is referenced by: frghash2spot 26590

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