Theorem fusgreg2wsplem 30420

Description: Lemma for fusgreg2wsp 30423 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 2749	. . . . 5 ⊢ (𝑎 = 𝑁 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑁))
2	1	rabbidv 3408	. . . 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 7401	. . . . 5 ⊢ (2 WSPathsN 𝐺) ∈ V
5	4	rabex 5286	. . . 4 ⊢ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ∈ V
6	2, 3, 5	fvmpt 6949	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑀‘𝑁) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁})
7	6	eleq2d 2823	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ 𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁}))
8		fveq1 6841	. . . 4 ⊢ (𝑤 = 𝑝 → (𝑤‘1) = (𝑝‘1))
9	8	eqeq1d 2739	. . 3 ⊢ (𝑤 = 𝑝 → ((𝑤‘1) = 𝑁 ↔ (𝑝‘1) = 𝑁))
10	9	elrab 3648	. 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401   ↦ cmpt 5181  ‘cfv 6500  (class class class)co 7368  1c1 11039  2c2 12212  Vtxcvtx 29081   WSPathsN cwwspthsn 29913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371

This theorem is referenced by:  fusgr2wsp2nb  30421  fusgreg2wsp  30423  2wspmdisj  30424