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Theorem membpartlem19 38767
Description: Together with disjlem19 38757, this is former prtlem19 38834. (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 38726 . . . 4 ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2 n0el2 38289 . . . . . . . 8 (¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)
32biimpi 216 . . . . . . 7 (¬ ∅ ∈ 𝐴 → dom ( E ↾ 𝐴) = 𝐴)
43ad2antll 728 . . . . . 6 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → dom ( E ↾ 𝐴) = 𝐴)
54eleq2d 2830 . . . . 5 ((𝐵𝑉 ∧ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) → (𝑢 ∈ dom ( E ↾ 𝐴) ↔ 𝑢𝐴))
6 eldisjlem19 38766 . . . . . . . 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 1537  wcel 2108  c0 4352   E cep 5598  ccnv 5699  dom cdm 5700  cres 5702  [cec 8761  ccoels 38136   ElDisj weldisj 38171   MembPart wmembpart 38176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765  df-qs 8769  df-coss 38367  df-coels 38368  df-cnvrefrel 38483  df-dmqs 38595  df-disjALTV 38661  df-eldisj 38663  df-part 38722  df-membpart 38724
This theorem is referenced by: (None)
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