Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjlem19 Structured version   Visualization version   GIF version

Theorem disjlem19 37195
Description: Lemma for disjdmqseq 37199, partim2 37201 and petlem 37206 via disjdmqs 37198, (general version of the former prtlem19 37272). (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 37194 . . . . . 6 ((𝐴𝑉𝑧 ∈ V) → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅𝐴𝑅𝑧))))
21elvd 3451 . . . . 5 (𝐴𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑧 ∈ [𝑥]𝑅𝐴𝑅𝑧))))
32imp31 419 . . . 4 (((𝐴𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (𝑧 ∈ [𝑥]𝑅𝐴𝑅𝑧))
4 elecALTV 36658 . . . . . 6 ((𝐴𝑉𝑧 ∈ V) → (𝑧 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝑧))
54elvd 3451 . . . . 5 (𝐴𝑉 → (𝑧 ∈ [𝐴] ≀ 𝑅𝐴𝑅𝑧))
65ad2antrr 725 . . . 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 2736 . 2 (((𝐴𝑉 ∧ Disj 𝑅) ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)
98exp31 421 1 (𝐴𝑉 → ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → [𝑥]𝑅 = [𝐴] ≀ 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3444   class class class wbr 5104  dom cdm 5632  [cec 8605  ccoss 36566   Disj wdisjALTV 36600
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 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-rmo 3352  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8609  df-coss 36805  df-cnvrefrel 36921  df-disjALTV 37099
This theorem is referenced by:  disjdmqsss  37196  disjdmqscossss  37197  eldisjlem19  37204
  Copyright terms: Public domain W3C validator