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Theorem inecmo 34439
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 relelec 8029 . . . . . . 7 (Rel 𝑅 → (𝑧 ∈ [𝐵]𝑅𝐵𝑅𝑧))
2 relelec 8029 . . . . . . 7 (Rel 𝑅 → (𝑧 ∈ [𝐶]𝑅𝐶𝑅𝑧))
31, 2anbi12d 618 . . . . . 6 (Rel 𝑅 → ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) ↔ (𝐵𝑅𝑧𝐶𝑅𝑧)))
43imbi1d 332 . . . . 5 (Rel 𝑅 → (((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ((𝐵𝑅𝑧𝐶𝑅𝑧) → 𝑥 = 𝑦)))
542ralbidv 3188 . . . 4 (Rel 𝑅 → (∀𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐵𝑅𝑧𝐶𝑅𝑧) → 𝑥 = 𝑦)))
6 inecmo.1 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
76breq1d 4865 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑅𝑧𝐶𝑅𝑧))
87rmo4 3608 . . . 4 (∃*𝑥𝐴 𝐵𝑅𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝐵𝑅𝑧𝐶𝑅𝑧) → 𝑥 = 𝑦))
95, 8syl6rbbr 281 . . 3 (Rel 𝑅 → (∃*𝑥𝐴 𝐵𝑅𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
109albidv 2011 . 2 (Rel 𝑅 → (∀𝑧∃*𝑥𝐴 𝐵𝑅𝑧 ↔ ∀𝑧𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
11 ineleq 34438 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
12 ralcom4 3429 . . 3 (∀𝑥𝐴𝑧𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑧𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
1311, 12bitri 266 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
1410, 13syl6rbbr 281 1 (Rel 𝑅 → (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥𝐴 𝐵𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 865  wal 1635   = wceq 1637  wcel 2157  wral 3107  ∃*wrmo 3110  cin 3779  c0 4127   class class class wbr 4855  Rel wrel 5327  [cec 7984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4986  ax-nul 4994  ax-pr 5107
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-br 4856  df-opab 4918  df-xp 5328  df-rel 5329  df-cnv 5330  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-ec 7988
This theorem is referenced by:  inecmo2  34440  ineccnvmo  34441
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