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Theorem fusgreg2wsplem 27515
Description: Lemma for fusgreg2wsp 27518 and related theorems. (Contributed by AV, 8-Jan-2022.)
Hypotheses
Ref Expression
frgrhash2wsp.v 𝑉 = (Vtx‘𝐺)
fusgreg2wsp.m 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
Assertion
Ref Expression
fusgreg2wsplem (𝑁𝑉 → (𝑝 ∈ (𝑀𝑁) ↔ (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
StepHypRef Expression
1 eqeq2 2782 . . . . 5 (𝑎 = 𝑁 → ((𝑤‘1) = 𝑎 ↔ (𝑤‘1) = 𝑁))
21rabbidv 3339 . . . 4 (𝑎 = 𝑁 → {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎} = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁})
3 fusgreg2wsp.m . . . 4 𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})
4 ovex 6827 . . . . 5 (2 WSPathsN 𝐺) ∈ V
54rabex 4947 . . . 4 {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ∈ V
62, 3, 5fvmpt 6426 . . 3 (𝑁𝑉 → (𝑀𝑁) = {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁})
76eleq2d 2836 . 2 (𝑁𝑉 → (𝑝 ∈ (𝑀𝑁) ↔ 𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁}))
8 fveq1 6332 . . . 4 (𝑤 = 𝑝 → (𝑤‘1) = (𝑝‘1))
98eqeq1d 2773 . . 3 (𝑤 = 𝑝 → ((𝑤‘1) = 𝑁 ↔ (𝑝‘1) = 𝑁))
109elrab 3515 . 2 (𝑝 ∈ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑁} ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁))
117, 10syl6bb 276 1 (𝑁𝑉 → (𝑝 ∈ (𝑀𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {crab 3065  cmpt 4864  cfv 6030  (class class class)co 6796  1c1 10143  2c2 11276  Vtxcvtx 26095   WSPathsN cwwspthsn 26956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ov 6799
This theorem is referenced by:  fusgr2wsp2nb  27516  fusgreg2wsp  27518  2wspmdisj  27519
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