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 38792
Description: Together with disjlem19 38782, this is former prtlem19 38859. (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 38751 . . . 4 ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2 n0el2 38314 . . . . . . . 8 (¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)
32biimpi 216 . . . . . . 7 (¬ ∅ ∈ 𝐴 → dom ( E ↾ 𝐴) = 𝐴)
43ad2antll 729 . . . . . 6 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → dom ( E ↾ 𝐴) = 𝐴)
54eleq2d 2824 . . . . 5 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom ( E ↾ 𝐴) ↔ 𝑢𝐴))
6 eldisjlem19 38791 . . . . . . . 8 (𝐵𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
76adantrd 491 . . . . . . 7 (𝐵𝑉 → (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
87imp 406 . . . . . 6 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
98expd 415 . . . . 5 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom ( E ↾ 𝐴) → (𝐵𝑢𝑢 = [𝐵] ∼ 𝐴)))
105, 9sylbird 260 . . . 4 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢𝐴 → (𝐵𝑢𝑢 = [𝐵] ∼ 𝐴)))
111, 10sylan2b 594 . . 3 ((𝐵𝑉 ∧ MembPart 𝐴) → (𝑢𝐴 → (𝐵𝑢𝑢 = [𝐵] ∼ 𝐴)))
1211impd 410 . 2 ((𝐵𝑉 ∧ MembPart 𝐴) → ((𝑢𝐴𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
1312ex 412 1 (𝐵𝑉 → ( MembPart 𝐴 → ((𝑢𝐴𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  c0 4338   E cep 5587  ccnv 5687  dom cdm 5688  cres 5690  [cec 8741  ccoels 38162   ElDisj weldisj 38197   MembPart wmembpart 38202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ec 8745  df-qs 8749  df-coss 38392  df-coels 38393  df-cnvrefrel 38508  df-dmqs 38620  df-disjALTV 38686  df-eldisj 38688  df-part 38747  df-membpart 38749
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator