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Theorem disjlem19 38782
Description: Lemma for disjdmqseq 38786, partim2 38788 and petlem 38793 via disjdmqs 38785, (general version of the former prtlem19 38859). (Contributed by Peter Mazsa, 16-Sep-2021.)
Assertion
Ref Expression
disjlem19 (𝐴𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅   𝑥,𝑉

Proof of Theorem disjlem19
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 disjlem18 38781 . . . . . 6 ((𝐴𝑉𝑧 ∈ V) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅𝐴𝑅𝑧))))
21elvd 3483 . . . . 5 (𝐴𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅𝐴𝑅𝑧))))
32imp31 417 . . . 4 (((𝐴𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝑥]𝑅𝐴𝑅𝑧))
4 elecALTV 38247 . . . . . 6 ((𝐴𝑉𝑧 ∈ V) → (𝑧 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝑧))
54elvd 3483 . . . . 5 (𝐴𝑉 → (𝑧 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝑧))
65ad2antrr 726 . . . 4 (((𝐴𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝑧))
73, 6bitr4d 282 . . 3 (((𝐴𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝑥]𝑅𝑧 ∈ [𝐴] ≀ 𝑅))
87eqrdv 2732 . 2 (((𝐴𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)
98exp31 419 1 (𝐴𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477   class class class wbr 5147  dom cdm 5688  [cec 8741  ccoss 38161   Disj wdisjALTV 38195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rmo 3377  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ec 8745  df-coss 38392  df-cnvrefrel 38508  df-disjALTV 38686
This theorem is referenced by:  disjdmqsss  38783  disjdmqscossss  38784  eldisjlem19  38791
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