Theorem fusgreg2wsplem 30943

Description: Lemma for fusgreg2wsp 30946 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 2772	. . . . 5 ⊢ (𝑎 = 𝑁 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑁))
2	1	rabbidv 3419	. . . 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 7457	. . . . 5 ⊢ (2 WSPathsN 𝐺) ∈ V
5	4	rabex 5300	. . . 4 ⊢ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ∈ V
6	2, 3, 5	fvmpt 6998	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑀‘𝑁) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁})
7	6	eleq2d 2846	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ 𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁}))
8		fveq1 6889	. . . 4 ⊢ (𝑤 = 𝑝 → (𝑤‘1) = (𝑝‘1))
9	8	eqeq1d 2762	. . 3 ⊢ (𝑤 = 𝑝 → ((𝑤‘1) = 𝑁 ↔ (𝑝‘1) = 𝑁))
10	9	elrab 3644	. 2 ⊢ (𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁))
11	7, 10	bitrdi 290	1 ⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412   ↦ cmpt 5185  ‘cfv 6543  (class class class)co 7424  1c1 11208  2c2 12404  Vtxcvtx 29583   WSPathsN cwwspthsn 30426

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427

This theorem is used by:  fusgr2wsp2nb  30944  fusgreg2wsp  30946  2wspmdisj  30947