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Theorem fusgreg2wsplem 29277
Description: Lemma for fusgreg2wsp 29280 and related theorems. (Contributed by AV, 8-Jan-2022.)
Hypotheses
Ref Expression
frgrhash2wsp.v 𝑉 = (Vtx‘𝐺)
fusgreg2wsp.m 𝑀 = (𝑎 ∈ 𝑉 ↩ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘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   𝑀,𝐺   𝑁,𝑎,𝑀   𝑀,𝑝
Allowed substitution hints:   𝐺(𝑝)   𝑀(𝑀,𝑝,𝑎)   𝑁(𝑝)   𝑉(𝑀,𝑝)

Proof of Theorem fusgreg2wsplem
StepHypRef Expression
1 eqeq2 2748 . . . . 5 (𝑎 = 𝑁 → ((𝑀‘1) = 𝑎 ↔ (𝑀‘1) = 𝑁))
21rabbidv 3415 . . . 4 (𝑎 = 𝑁 → {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑎} = {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑁})
3 fusgreg2wsp.m . . . 4 𝑀 = (𝑎 ∈ 𝑉 ↩ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑎})
4 ovex 7390 . . . . 5 (2 WSPathsN 𝐺) ∈ V
54rabex 5289 . . . 4 {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑁} ∈ V
62, 3, 5fvmpt 6948 . . 3 (𝑁 ∈ 𝑉 → (𝑀‘𝑁) = {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑁})
76eleq2d 2823 . 2 (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ 𝑝 ∈ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑁}))
8 fveq1 6841 . . . 4 (𝑀 = 𝑝 → (𝑀‘1) = (𝑝‘1))
98eqeq1d 2738 . . 3 (𝑀 = 𝑝 → ((𝑀‘1) = 𝑁 ↔ (𝑝‘1) = 𝑁))
109elrab 3645 . 2 (𝑝 ∈ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑁} ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁))
117, 10bitrdi 286 1 (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁)))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3407   ↩ cmpt 5188  â€˜cfv 6496  (class class class)co 7357  1c1 11052  2c2 12208  Vtxcvtx 27947   WSPathsN cwwspthsn 28773
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360
This theorem is referenced by:  fusgr2wsp2nb  29278  fusgreg2wsp  29280  2wspmdisj  29281
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