Theorem fusgreg2wsplem 28697

Description: Lemma for fusgreg2wsp 28700 and related theorems. (Contributed by AV, 8-Jan-2022.)

Hypotheses

Ref	Expression
frgrhash2wsp.v	⊢ 𝑉 = (Vtx‘𝐺)
fusgreg2wsp.m	⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})

Assertion

Ref	Expression
fusgreg2wsplem	⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁)))

Distinct variable groups:   𝐺,𝑎   𝑉,𝑎   𝑤,𝐺   𝑁,𝑎,𝑤   𝑤,𝑝

Allowed substitution hints:   𝐺(𝑝)   𝑀(𝑤,𝑝,𝑎)   𝑁(𝑝)   𝑉(𝑤,𝑝)

Proof of Theorem fusgreg2wsplem

Step	Hyp	Ref	Expression
1		eqeq2 2750	. . . . 5 ⊢ (𝑎 = 𝑁 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑁))
2	1	rabbidv 3414	. . . 4 ⊢ (𝑎 = 𝑁 → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎} = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁})
3		fusgreg2wsp.m	. . . 4 ⊢ 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
4		ovex 7308	. . . . 5 ⊢ (2 WSPathsN 𝐺) ∈ V
5	4	rabex 5256	. . . 4 ⊢ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ∈ V
6	2, 3, 5	fvmpt 6875	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑀‘𝑁) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁})
7	6	eleq2d 2824	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ 𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁}))
8		fveq1 6773	. . . 4 ⊢ (𝑤 = 𝑝 → (𝑤‘1) = (𝑝‘1))
9	8	eqeq1d 2740	. . 3 ⊢ (𝑤 = 𝑝 → ((𝑤‘1) = 𝑁 ↔ (𝑝‘1) = 𝑁))
10	9	elrab 3624	. 2 ⊢ (𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁))
11	7, 10	bitrdi 287	1 ⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  1c1 10872  2c2 12028  Vtxcvtx 27366   WSPathsN cwwspthsn 28193

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-sep 5223  ax-nul 5230  ax-pr 5352

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-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  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-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278

This theorem is referenced by:  fusgr2wsp2nb  28698  fusgreg2wsp  28700  2wspmdisj  28701