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Theorem eldisjlem19 37772
Description: Special case of disjlem19 37763 (together with membpartlem19 37773, this is former prtlem19 37840). (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 37669 . . . . . . . 8 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
2 disjlem19 37763 . . . . . . . 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 407 . . . . . 6 ((𝐵𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵 ∈ [𝑢]( E ↾ 𝐴)) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)))
54expdimp 453 . . . . 5 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → (𝐵 ∈ [𝑢]( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = [𝐵] ≀ ( E ↾ 𝐴)))
6 eccnvepres3 37246 . . . . . . . 8 (𝑢 ∈ dom ( E ↾ 𝐴) → [𝑢]( E ↾ 𝐴) = 𝑢)
76eleq2d 2819 . . . . . . 7 (𝑢 ∈ dom ( E ↾ 𝐴) → (𝐵 ∈ [𝑢]( E ↾ 𝐴) ↔ 𝐵𝑢))
86eqeq1d 2734 . . . . . . 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 482 . . . . 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 37374 . . . . . 6 𝐴 = ≀ ( E ↾ 𝐴)
1312eceq2i 8746 . . . . 5 [𝐵] ∼ 𝐴 = [𝐵] ≀ ( E ↾ 𝐴)
1413eqeq2i 2745 . . . 4 (𝑢 = [𝐵] ∼ 𝐴𝑢 = [𝐵] ≀ ( E ↾ 𝐴))
1511, 14imbitrrdi 251 . . 3 (((𝐵𝑉 ∧ ElDisj 𝐴) ∧ 𝑢 ∈ dom ( E ↾ 𝐴)) → (𝐵𝑢𝑢 = [𝐵] ∼ 𝐴))
1615expimpd 454 . 2 ((𝐵𝑉 ∧ ElDisj 𝐴) → ((𝑢 ∈ dom ( E ↾ 𝐴) ∧ 𝐵𝑢) → 𝑢 = [𝐵] ∼ 𝐴))
1716ex 413 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 396   = wceq 1541  wcel 2106   E cep 5579  ccnv 5675  dom cdm 5676  cres 5678  [cec 8703  ccoss 37135  ccoels 37136   Disj wdisjALTV 37169   ElDisj weldisj 37171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-eprel 5580  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8707  df-coss 37373  df-coels 37374  df-cnvrefrel 37489  df-disjALTV 37667  df-eldisj 37669
This theorem is referenced by:  membpartlem19  37773
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