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Theorem membpartlem19 38812
Description: Together with disjlem19 38802, this is former prtlem19 38879. (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 38771 . . . 4 ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2 n0el2 38334 . . . . . . . 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 2827 . . . . 5 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom ( E ↾ 𝐴) ↔ 𝑢𝐴))
6 eldisjlem19 38811 . . . . . . . 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 1540  wcel 2108  c0 4333   E cep 5583  ccnv 5684  dom cdm 5685  cres 5687  [cec 8743  ccoels 38183   ElDisj weldisj 38218   MembPart wmembpart 38223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751  df-coss 38412  df-coels 38413  df-cnvrefrel 38528  df-dmqs 38640  df-disjALTV 38706  df-eldisj 38708  df-part 38767  df-membpart 38769
This theorem is referenced by: (None)
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