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Theorem eldisjlem19 37680
Description: Special case of disjlem19 37671 (together with membpartlem19 37681, this is former prtlem19 37748). (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 37577 . . . . . . . 8 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
2 disjlem19 37671 . . . . . . . 8 (𝐵𝑉 → ( Disj ( E ↾ 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢]( E ↾ 𝐴)) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴))))
31, 2biimtrid 241 . . . . . . 7 (𝐵𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢]( E ↾ 𝐴)) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴))))
43imp 408 . . . . . 6 ((𝐵𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢]( E ↾ 𝐴)) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)))
54expdimp 454 . . . . 5 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → (𝐵 ∈ [𝑢]( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)))
6 eccnvepres3 37154 . . . . . . . 8 (𝑢 ∈ dom ( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = 𝑢)
76eleq2d 2820 . . . . . . 7 (𝑢 ∈ dom ( E ↾ 𝐴) → (𝐵 ∈ [𝑢]( E ↾ 𝐴) ↔ 𝐵𝑢))
86eqeq1d 2735 . . . . . . 7 (𝑢 ∈ dom ( E ↾ 𝐴) → ([𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴) ↔ 𝑢 = [𝐵] ≀ ( E ↾ 𝐴)))
97, 8imbi12d 345 . . . . . 6 (𝑢 ∈ dom ( E ↾ 𝐴) → ((𝐵 ∈ [𝑢]( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)) ↔ (𝐵𝑢𝑢 = [𝐵] ≀ ( E ↾ 𝐴))))
109adantl 483 . . . . 5 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → ((𝐵 ∈ [𝑢]( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)) ↔ (𝐵𝑢𝑢 = [𝐵] ≀ ( E ↾ 𝐴))))
115, 10mpbid 231 . . . 4 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → (𝐵𝑢𝑢 = [𝐵] ≀ ( E ↾ 𝐴)))
12 df-coels 37282 . . . . . 6 𝐴 = ≀ ( E ↾ 𝐴)
1312eceq2i 8744 . . . . 5 [𝐵] ∼ 𝐴 = [𝐵] ≀ ( E ↾ 𝐴)
1413eqeq2i 2746 . . . 4 (𝑢 = [𝐵] ∼ 𝐴𝑢 = [𝐵] ≀ ( E ↾ 𝐴))
1511, 14syl6ibr 252 . . 3 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → (𝐵𝑢𝑢 = [𝐵] ∼ 𝐴))
1615expimpd 455 . 2 ((𝐵𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
1716ex 414 1 (𝐵𝑉 → ( ElDisj 𝐴 → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   E cep 5580  ccnv 5676  dom cdm 5677  cres 5679  [cec 8701  ccoss 37043  ccoels 37044   Disj wdisjALTV 37077   ElDisj weldisj 37079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8705  df-coss 37281  df-coels 37282  df-cnvrefrel 37397  df-disjALTV 37575  df-eldisj 37577
This theorem is referenced by:  membpartlem19  37681
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