Theorem fusgreg2wsplem 27197

Description: Lemma for fusgreg2wsp 27200 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 2633	. . . . 5 ⊢ (𝑎 = 𝑁 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑁))
2	1	rabbidv 3189	. . . 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 6678	. . . . 5 ⊢ (2 WSPathsN 𝐺) ∈ V
5	4	rabex 4813	. . . 4 ⊢ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ∈ V
6	2, 3, 5	fvmpt 6282	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝑀‘𝑁) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁})
7	6	eleq2d 2687	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ 𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁}))
8		fveq1 6190	. . . 4 ⊢ (𝑤 = 𝑝 → (𝑤‘1) = (𝑝‘1))
9	8	eqeq1d 2624	. . 3 ⊢ (𝑤 = 𝑝 → ((𝑤‘1) = 𝑁 ↔ (𝑝‘1) = 𝑁))
10	9	elrab 3363	. 2 ⊢ (𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁))
11	7, 10	syl6bb 276	1 ⊢ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916   ↦ cmpt 4729  ‘cfv 5888  (class class class)co 6650  1c1 9937  2c2 11070  Vtxcvtx 25874   WSPathsN cwwspthsn 26720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  fusgr2wsp2nb  27198  fusgreg2wsp  27200  2wspmdisj  27201