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Theorem spthonisspth 28118

Description: A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 18-Jan-2021.)

**Assertion**
Ref	Expression
spthonisspth	⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → 𝐹(SPaths‘𝐺)𝑃)

**Proof of Theorem spthonisspth**
Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	spthonprop 28113	. 2 ⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃)))
3		simp3r 1201	. 2 ⊢ (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃)) → 𝐹(SPaths‘𝐺)𝑃)
4	2, 3	syl 17	1 ⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → 𝐹(SPaths‘𝐺)𝑃)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2106 Vcvv 3432 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Vtxcvtx 27366 TrailsOnctrlson 28059 SPathscspths 28081 SPathsOncspthson 28083

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-spthson 28087

This theorem is referenced by: usgr2trlspth 28129 wspthsnonn0vne 28282

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