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Theorem disjlem18 37756
Description: Lemma for disjdmqseq 37761, partim2 37763 and petlem 37768 via disjlem19 37757, (general version of the former prtlem18 37833). (Contributed by Peter Mazsa, 16-Sep-2021.)
Assertion
Ref Expression
disjlem18 ((𝐴𝑉𝐵𝑊) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅𝐴𝑅𝐵))))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑉   𝑥,𝑊

Proof of Theorem disjlem18
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rspe 3246 . . . . . . 7 ((𝑥 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)) → ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅))
21expr 457 . . . . . 6 ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅 → ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
32adantl 482 . . . . 5 ((((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (𝐵 ∈ [𝑥]𝑅 → ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
4 relbrcoss 37402 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → (Rel 𝑅 → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅))))
5 disjrel 37686 . . . . . . 7 ( Disj 𝑅 → Rel 𝑅)
64, 5impel 506 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
76adantr 481 . . . . 5 ((((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (𝐴𝑅𝐵 ↔ ∃𝑥 ∈ dom 𝑅(𝐴 ∈ [𝑥]𝑅𝐵 ∈ [𝑥]𝑅)))
83, 7sylibrd 258 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (𝐵 ∈ [𝑥]𝑅𝐴𝑅𝐵))
98ex 413 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅𝐴𝑅𝐵)))
10 disjlem17 37755 . . . . 5 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
1110adantl 482 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
12 relbrcoss 37402 . . . . . 6 ((𝐴𝑉𝐵𝑊) → (Rel 𝑅 → (𝐴𝑅𝐵 ↔ ∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅))))
1312, 5impel 506 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) → (𝐴𝑅𝐵 ↔ ∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)))
1413imbi1d 341 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) → ((𝐴𝑅𝐵𝐵 ∈ [𝑥]𝑅) ↔ (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
1511, 14sylibrd 258 . . 3 (((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝐴𝑅𝐵𝐵 ∈ [𝑥]𝑅)))
169, 15impbidd 209 . 2 (((𝐴𝑉𝐵𝑊) ∧ Disj 𝑅) → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅𝐴𝑅𝐵)))
1716ex 413 1 ((𝐴𝑉𝐵𝑊) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝐵 ∈ [𝑥]𝑅𝐴𝑅𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  wrex 3070   class class class wbr 5148  dom cdm 5676  Rel wrel 5681  [cec 8703  ccoss 37129   Disj wdisjALTV 37163
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-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-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 37367  df-cnvrefrel 37483  df-disjALTV 37661
This theorem is referenced by:  disjlem19  37757
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