Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjlem17 Structured version   Visualization version   GIF version

Theorem disjlem17 38800
Description: Lemma for disjdmqseq 38806, partim2 38808 and petlem 38813 via disjlem18 38801, (general version of the former prtlem17 38877). (Contributed by Peter Mazsa, 10-Sep-2021.)
Assertion
Ref Expression
disjlem17 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem disjlem17
StepHypRef Expression
1 df-rex 3071 . . 3 (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) ↔ ∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)))
2 an32 646 . . . . . . . 8 (((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) ↔ ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅))
3 disjlem14 38799 . . . . . . . . . . 11 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
4 eleq2 2830 . . . . . . . . . . . 12 ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑥]𝑅𝐵 ∈ [𝑦]𝑅))
54biimprd 248 . . . . . . . . . . 11 ([𝑥]𝑅 = [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))
63, 5syl8 76 . . . . . . . . . 10 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))))
76exp4a 431 . . . . . . . . 9 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑥]𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅)))))
87impd 410 . . . . . . . 8 ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) ∧ 𝐴 ∈ [𝑥]𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))))
92, 8biimtrrid 243 . . . . . . 7 ( Disj 𝑅 → (((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) ∧ 𝑦 ∈ dom 𝑅) → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅))))
109expd 415 . . . . . 6 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → (𝐴 ∈ [𝑦]𝑅 → (𝐵 ∈ [𝑦]𝑅𝐵 ∈ [𝑥]𝑅)))))
1110imp5a 440 . . . . 5 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (𝑦 ∈ dom 𝑅 → ((𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))))
1211imp4b 421 . . . 4 (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → ((𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
1312exlimdv 1933 . . 3 (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (∃𝑦(𝑦 ∈ dom 𝑅 ∧ (𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅)) → 𝐵 ∈ [𝑥]𝑅))
141, 13biimtrid 242 . 2 (( Disj 𝑅 ∧ (𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅)) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅))
1514ex 412 1 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝐴 ∈ [𝑥]𝑅) → (∃𝑦 ∈ dom 𝑅(𝐴 ∈ [𝑦]𝑅𝐵 ∈ [𝑦]𝑅) → 𝐵 ∈ [𝑥]𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  dom cdm 5685  [cec 8743   Disj wdisjALTV 38216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-coss 38412  df-cnvrefrel 38528  df-disjALTV 38706
This theorem is referenced by:  disjlem18  38801
  Copyright terms: Public domain W3C validator