Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  membpartlem19 Structured version   Visualization version   GIF version

Theorem membpartlem19 38335
Description: Together with disjlem19 38325, this is former prtlem19 38402. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.) (Revised by Peter Mazsa, 21-Oct-2021.)
Assertion
Ref Expression
membpartlem19 (𝐵𝑉 → ( MembPart 𝐴 → ((𝑢𝐴𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝑉

Proof of Theorem membpartlem19
StepHypRef Expression
1 dfmembpart2 38294 . . . 4 ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2 n0el2 37857 . . . . . . . 8 (¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)
32biimpi 215 . . . . . . 7 (¬ ∅ ∈ 𝐴 → dom ( E ↾ 𝐴) = 𝐴)
43ad2antll 727 . . . . . 6 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → dom ( E ↾ 𝐴) = 𝐴)
54eleq2d 2811 . . . . 5 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom ( E ↾ 𝐴) ↔ 𝑢𝐴))
6 eldisjlem19 38334 . . . . . . . 8 (𝐵𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
76adantrd 490 . . . . . . 7 (𝐵𝑉 → (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
87imp 405 . . . . . 6 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
98expd 414 . . . . 5 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom ( E ↾ 𝐴) → (𝐵𝑢𝑢 = [𝐵] ∼ 𝐴)))
105, 9sylbird 259 . . . 4 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢𝐴 → (𝐵𝑢𝑢 = [𝐵] ∼ 𝐴)))
111, 10sylan2b 592 . . 3 ((𝐵𝑉 ∧ MembPart 𝐴) → (𝑢𝐴 → (𝐵𝑢𝑢 = [𝐵] ∼ 𝐴)))
1211impd 409 . 2 ((𝐵𝑉 ∧ MembPart 𝐴) → ((𝑢𝐴𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
1312ex 411 1 (𝐵𝑉 → ( MembPart 𝐴 → ((𝑢𝐴𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  c0 4319   E cep 5576  ccnv 5672  dom cdm 5673  cres 5675  [cec 8716  ccoels 37702   ElDisj weldisj 37737   MembPart wmembpart 37742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-id 5571  df-eprel 5577  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ec 8720  df-qs 8724  df-coss 37935  df-coels 37936  df-cnvrefrel 38051  df-dmqs 38163  df-disjALTV 38229  df-eldisj 38231  df-part 38290  df-membpart 38292
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator