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Theorem eldisjlem19 38811
Description: Special case of disjlem19 38802 (together with membpartlem19 38812, this is former prtlem19 38879). (Contributed by Peter Mazsa, 21-Oct-2021.)
Assertion
Ref Expression
eldisjlem19 (𝐵𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝑉

Proof of Theorem eldisjlem19
StepHypRef Expression
1 df-eldisj 38708 . . . . . . . 8 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
2 disjlem19 38802 . . . . . . . 8 (𝐵𝑉 → ( Disj ( E ↾ 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢]( E ↾ 𝐴)) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴))))
31, 2biimtrid 242 . . . . . . 7 (𝐵𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢]( E ↾ 𝐴)) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴))))
43imp 406 . . . . . 6 ((𝐵𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢]( E ↾ 𝐴)) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)))
54expdimp 452 . . . . 5 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → (𝐵 ∈ [𝑢]( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)))
6 eccnvepres3 38287 . . . . . . . 8 (𝑢 ∈ dom ( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = 𝑢)
76eleq2d 2827 . . . . . . 7 (𝑢 ∈ dom ( E ↾ 𝐴) → (𝐵 ∈ [𝑢]( E ↾ 𝐴) ↔ 𝐵𝑢))
86eqeq1d 2739 . . . . . . 7 (𝑢 ∈ dom ( E ↾ 𝐴) → ([𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴) ↔ 𝑢 = [𝐵] ≀ ( E ↾ 𝐴)))
97, 8imbi12d 344 . . . . . 6 (𝑢 ∈ dom ( E ↾ 𝐴) → ((𝐵 ∈ [𝑢]( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)) ↔ (𝐵𝑢𝑢 = [𝐵] ≀ ( E ↾ 𝐴))))
109adantl 481 . . . . 5 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → ((𝐵 ∈ [𝑢]( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)) ↔ (𝐵𝑢𝑢 = [𝐵] ≀ ( E ↾ 𝐴))))
115, 10mpbid 232 . . . 4 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → (𝐵𝑢𝑢 = [𝐵] ≀ ( E ↾ 𝐴)))
12 df-coels 38413 . . . . . 6 𝐴 = ≀ ( E ↾ 𝐴)
1312eceq2i 8787 . . . . 5 [𝐵] ∼ 𝐴 = [𝐵] ≀ ( E ↾ 𝐴)
1413eqeq2i 2750 . . . 4 (𝑢 = [𝐵] ∼ 𝐴𝑢 = [𝐵] ≀ ( E ↾ 𝐴))
1511, 14imbitrrdi 252 . . 3 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → (𝐵𝑢𝑢 = [𝐵] ∼ 𝐴))
1615expimpd 453 . 2 ((𝐵𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
1716ex 412 1 (𝐵𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108   E cep 5583  ccnv 5684  dom cdm 5685  cres 5687  [cec 8743  ccoss 38182  ccoels 38183   Disj wdisjALTV 38216   ElDisj weldisj 38218
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-coss 38412  df-coels 38413  df-cnvrefrel 38528  df-disjALTV 38706  df-eldisj 38708
This theorem is referenced by:  membpartlem19  38812
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