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Theorem membpartlem19 38207
Description: Together with disjlem19 38197, this is former prtlem19 38274. (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 38166 . . . 4 ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2 n0el2 37728 . . . . . . . 8 (¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)
32biimpi 215 . . . . . . 7 (¬ ∅ ∈ 𝐴 → dom ( E ↾ 𝐴) = 𝐴)
43ad2antll 728 . . . . . 6 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → dom ( E ↾ 𝐴) = 𝐴)
54eleq2d 2814 . . . . 5 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom ( E ↾ 𝐴) ↔ 𝑢𝐴))
6 eldisjlem19 38206 . . . . . . . 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 593 . . 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 1534  wcel 2099  c0 4318   E cep 5575  ccnv 5671  dom cdm 5672  cres 5674  [cec 8714  ccoels 37571   ElDisj weldisj 37606   MembPart wmembpart 37611
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-rmo 3371  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-br 5143  df-opab 5205  df-id 5570  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8718  df-qs 8722  df-coss 37807  df-coels 37808  df-cnvrefrel 37923  df-dmqs 38035  df-disjALTV 38101  df-eldisj 38103  df-part 38162  df-membpart 38164
This theorem is referenced by: (None)
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