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Theorem 2spthonot 26393

Description: The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)

**Assertion**
Ref	Expression
2spthonot	⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (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))})

Distinct variable groups: 𝑡,𝐸,𝑓,𝑝 𝑡,𝑉,𝑓,𝑝 𝐴,𝑓,𝑝,𝑡 𝐵,𝑓,𝑝,𝑡 Allowed substitution hints: 𝑋(𝑡,𝑓,𝑝) 𝑌(𝑡,𝑓,𝑝)

Proof of Theorem 2spthonot Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		is2spthonot 26391	. . . 4 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 2SPathOnOt 𝐸) = (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))}))
2	1	adantr 480	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑉 2SPathOnOt 𝐸) = (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))}))
3	2	oveqd 6566	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = (𝐴(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))})𝐵))
4		simprl 790	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑉)
5		simprr 792	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉)
6		3xpexg 6859	. . . . 5 ⊢ (𝑉 ∈ 𝑋 → ((𝑉 × 𝑉) × 𝑉) ∈ V)
7	6	ad2antrr 758	. . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → ((𝑉 × 𝑉) × 𝑉) ∈ V)
8		rabexg 4739	. . . 4 ⊢ (((𝑉 × 𝑉) × 𝑉) ∈ 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))} ∈ V)
9	7, 8	syl 17	. . 3 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (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))} ∈ V)
10		oveq12 6558	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑎(𝑉 SPathOn 𝐸)𝑏) = (𝐴(𝑉 SPathOn 𝐸)𝐵))
11	10	breqd 4594	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝑓(𝑎(𝑉 SPathOn 𝐸)𝑏)𝑝 ↔ 𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝))
12		eqeq2 2621	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((1^st ‘(1^st ‘𝑡)) = 𝑎 ↔ (1^st ‘(1^st ‘𝑡)) = 𝐴))
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((1^st ‘(1^st ‘𝑡)) = 𝑎 ↔ (1^st ‘(1^st ‘𝑡)) = 𝐴))
14		eqeq2 2621	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → ((2^nd ‘𝑡) = 𝑏 ↔ (2^nd ‘𝑡) = 𝐵))
15	14	adantl 481	. . . . . . . 8 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((2^nd ‘𝑡) = 𝑏 ↔ (2^nd ‘𝑡) = 𝐵))
16	13, 15	3anbi13d 1393	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((1^st ‘(1^st ‘𝑡)) = 𝑎 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝑏) ↔ ((1^st ‘(1^st ‘𝑡)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑡)) = (𝑝‘1) ∧ (2^nd ‘𝑡) = 𝐵)))
17	11, 16	3anbi13d 1393	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((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)) ↔ (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))))
18	17	2exbidv 1839	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (∃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)) ↔ ∃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))))
19	18	rabbidv 3164	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 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))} = {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))})
20		eqid 2610	. . . 4 ⊢ (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))}) = (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))})
21	19, 20	ovmpt2ga 6688	. . 3 ⊢ ((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))} ∈ V) → (𝐴(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))})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))})
22	4, 5, 9, 21	syl3anc 1318	. 2 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(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))})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))})
23	3, 22	eqtrd 2644	1 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (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))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {crab 2900 Vcvv 3173 class class class wbr 4583 × cxp 5036 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058 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-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-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 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-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-1st 7059 df-2nd 7060 df-2spthonot 26387

This theorem is referenced by: el2spthonot 26397 el2spthonot0 26398 2spthonot3v 26403 2pthwlkonot 26412 2spotfi 26419 2spotdisj 26588

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