isspth - Metamath Proof Explorer

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Theorem isspth 29742

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 29740	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ^◡𝑝)}
2		cnveq 5884	. . . 4 ⊢ (𝑝 = 𝑃 → ^◡𝑝 = ^◡𝑃)
3	2	funeqd 6588	. . 3 ⊢ (𝑝 = 𝑃 → (Fun ^◡𝑝 ↔ Fun ^◡𝑃))
4	3	adantl 481	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ^◡𝑝 ↔ Fun ^◡𝑃))
5		reltrls 29712	. 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 1540 class class class wbr 5143 ^◡ccnv 5684 Fun wfun 6555 ‘cfv 6561 Trailsctrls 29708 SPathscspths 29731

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-trls 29710 df-spths 29735

This theorem is referenced by: spthispth 29744 spthdifv 29753 spthdep 29754 pthdepisspth 29755 spthonepeq 29772 uhgrwkspth 29775 usgr2wlkspth 29779 usgr2pth 29784 2spthd 29961 0spth 30145 3spthd 30195 spthcycl 35134

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