MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fusgreg2wsplem Structured version   Visualization version   GIF version

Theorem fusgreg2wsplem 30130
Description: Lemma for fusgreg2wsp 30133 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 2739 . . . . 5 (𝑎 = 𝑁 → ((𝑀‘1) = 𝑎 ↔ (𝑀‘1) = 𝑁))
21rabbidv 3435 . . . 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 7447 . . . . 5 (2 WSPathsN 𝐺) ∈ V
54rabex 5328 . . . 4 {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑁} ∈ V
62, 3, 5fvmpt 6999 . . 3 (𝑁 ∈ 𝑉 → (𝑀‘𝑁) = {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑁})
76eleq2d 2814 . 2 (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ 𝑝 ∈ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑁}))
8 fveq1 6890 . . . 4 (𝑀 = 𝑝 → (𝑀‘1) = (𝑝‘1))
98eqeq1d 2729 . . 3 (𝑀 = 𝑝 → ((𝑀‘1) = 𝑁 ↔ (𝑝‘1) = 𝑁))
109elrab 3680 . 2 (𝑝 ∈ {𝑀 ∈ (2 WSPathsN 𝐺) ∣ (𝑀‘1) = 𝑁} ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁))
117, 10bitrdi 287 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 395   = wceq 1534   ∈ wcel 2099  {crab 3427   ↩ cmpt 5225  â€˜cfv 6542  (class class class)co 7414  1c1 11131  2c2 12289  Vtxcvtx 28796   WSPathsN cwwspthsn 29626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417
This theorem is referenced by:  fusgr2wsp2nb  30131  fusgreg2wsp  30133  2wspmdisj  30134
  Copyright terms: Public domain W3C validator