isspth - Metamath Proof Explorer

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Theorem isspth 29757

Description: Conditions for a pair of classes/functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)

**Assertion**
Ref	Expression
isspth	⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃))

Proof of Theorem isspth Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		spthsfval 29755	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ^◡𝑝)}
2		cnveq 5887	. . . 4 ⊢ (𝑝 = 𝑃 → ^◡𝑝 = ^◡𝑃)
3	2	funeqd 6590	. . 3 ⊢ (𝑝 = 𝑃 → (Fun ^◡𝑝 ↔ Fun ^◡𝑃))
4	3	adantl 481	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ^◡𝑝 ↔ Fun ^◡𝑃))
5		reltrls 29727	. 2 ⊢ Rel (Trails‘𝐺)
6	1, 4, 5	brfvopabrbr 7013	1 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 class class class wbr 5148 ^◡ccnv 5688 Fun wfun 6557 ‘cfv 6563 Trailsctrls 29723 SPathscspths 29746

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-trls 29725 df-spths 29750

This theorem is referenced by: spthispth 29759 spthdifv 29766 spthdep 29767 pthdepisspth 29768 spthonepeq 29785 uhgrwkspth 29788 usgr2wlkspth 29792 usgr2pth 29797 2spthd 29971 0spth 30155 3spthd 30205 spthcycl 35114

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