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Theorem inecmo 38357
Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 29-May-2018.)
Hypothesis
Ref Expression
inecmo.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
inecmo (Rel 𝑅 → (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥𝐴 𝐵𝑅𝑧))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑥,𝐶,𝑧   𝑥,𝑅,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem inecmo
StepHypRef Expression
1 ineleq 38356 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
2 ralcom4 3285 . . 3 (∀𝑥𝐴𝑧𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑧𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
31, 2bitri 275 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
4 inecmo.1 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
54breq1d 5152 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑅𝑧𝐶𝑅𝑧))
65rmo4 3735 . . . 4 (∃*𝑥𝐴 𝐵𝑅𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝐵𝑅𝑧𝐶𝑅𝑧) → 𝑥 = 𝑦))
7 relelec 8793 . . . . . . 7 (Rel 𝑅 → (𝑧 ∈ [𝐵]𝑅𝐵𝑅𝑧))
8 relelec 8793 . . . . . . 7 (Rel 𝑅 → (𝑧 ∈ [𝐶]𝑅𝐶𝑅𝑧))
97, 8anbi12d 632 . . . . . 6 (Rel 𝑅 → ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) ↔ (𝐵𝑅𝑧𝐶𝑅𝑧)))
109imbi1d 341 . . . . 5 (Rel 𝑅 → (((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ((𝐵𝑅𝑧𝐶𝑅𝑧) → 𝑥 = 𝑦)))
11102ralbidv 3220 . . . 4 (Rel 𝑅 → (∀𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐵𝑅𝑧𝐶𝑅𝑧) → 𝑥 = 𝑦)))
126, 11bitr4id 290 . . 3 (Rel 𝑅 → (∃*𝑥𝐴 𝐵𝑅𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
1312albidv 1919 . 2 (Rel 𝑅 → (∀𝑧∃*𝑥𝐴 𝐵𝑅𝑧 ↔ ∀𝑧𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
143, 13bitr4id 290 1 (Rel 𝑅 → (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥𝐴 𝐵𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  wal 1537   = wceq 1539  wcel 2107  wral 3060  ∃*wrmo 3378  cin 3949  c0 4332   class class class wbr 5142  Rel wrel 5689  [cec 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rmo 3379  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748
This theorem is referenced by:  inecmo2  38358  ineccnvmo  38359  disjres  38746
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