isspth - Metamath Proof Explorer

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Theorem isspth 29580

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 29578	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ^◡𝑝)}
2		cnveq 5870	. . . 4 ⊢ (𝑝 = 𝑃 → ^◡𝑝 = ^◡𝑃)
3	2	funeqd 6569	. . 3 ⊢ (𝑝 = 𝑃 → (Fun ^◡𝑝 ↔ Fun ^◡𝑃))
4	3	adantl 480	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ^◡𝑝 ↔ Fun ^◡𝑃))
5		reltrls 29550	. 2 ⊢ Rel (Trails‘𝐺)
6	1, 4, 5	brfvopabrbr 6996	1 ⊢ (𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ^◡𝑃))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 class class class wbr 5143 ^◡ccnv 5671 Fun wfun 6536 ‘cfv 6542 Trailsctrls 29546 SPathscspths 29569

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-trls 29548 df-spths 29573

This theorem is referenced by: spthispth 29582 spthdifv 29589 spthdep 29590 pthdepisspth 29591 spthonepeq 29608 uhgrwkspth 29611 usgr2wlkspth 29615 usgr2pth 29620 2spthd 29794 0spth 29978 3spthd 30028 spthcycl 34795

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