isspth - Metamath Proof Explorer

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Theorem isspth 29760

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 29758	. 2 ⊢ (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun ^◡𝑝)}
2		cnveq 5898	. . . 4 ⊢ (𝑝 = 𝑃 → ^◡𝑝 = ^◡𝑃)
3	2	funeqd 6600	. . 3 ⊢ (𝑝 = 𝑃 → (Fun ^◡𝑝 ↔ Fun ^◡𝑃))
4	3	adantl 481	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (Fun ^◡𝑝 ↔ Fun ^◡𝑃))
5		reltrls 29730	. 2 ⊢ Rel (Trails‘𝐺)
6	1, 4, 5	brfvopabrbr 7026	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 5166 ^◡ccnv 5699 Fun wfun 6567 ‘cfv 6573 Trailsctrls 29726 SPathscspths 29749

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fv 6581 df-trls 29728 df-spths 29753

This theorem is referenced by: spthispth 29762 spthdifv 29769 spthdep 29770 pthdepisspth 29771 spthonepeq 29788 uhgrwkspth 29791 usgr2wlkspth 29795 usgr2pth 29800 2spthd 29974 0spth 30158 3spthd 30208 spthcycl 35097

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