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Theorem inecmo 35768
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 8321 . . . . . . 7 (Rel 𝑅 → (𝑧 ∈ [𝐵]𝑅𝐵𝑅𝑧))
2 relelec 8321 . . . . . . 7 (Rel 𝑅 → (𝑧 ∈ [𝐶]𝑅𝐶𝑅𝑧))
31, 2anbi12d 633 . . . . . 6 (Rel 𝑅 → ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) ↔ (𝐵𝑅𝑧𝐶𝑅𝑧)))
43imbi1d 345 . . . . 5 (Rel 𝑅 → (((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ((𝐵𝑅𝑧𝐶𝑅𝑧) → 𝑥 = 𝑦)))
542ralbidv 3167 . . . 4 (Rel 𝑅 → (∀𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑥𝐴𝑦𝐴 ((𝐵𝑅𝑧𝐶𝑅𝑧) → 𝑥 = 𝑦)))
6 inecmo.1 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
76breq1d 5043 . . . . 5 (𝑥 = 𝑦 → (𝐵𝑅𝑧𝐶𝑅𝑧))
87rmo4 3672 . . . 4 (∃*𝑥𝐴 𝐵𝑅𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝐵𝑅𝑧𝐶𝑅𝑧) → 𝑥 = 𝑦))
95, 8syl6rbbr 293 . . 3 (Rel 𝑅 → (∃*𝑥𝐴 𝐵𝑅𝑧 ↔ ∀𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
109albidv 1921 . 2 (Rel 𝑅 → (∀𝑧∃*𝑥𝐴 𝐵𝑅𝑧 ↔ ∀𝑧𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
11 ineleq 35767 . . 3 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑥𝐴𝑧𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
12 ralcom4 3201 . . 3 (∀𝑥𝐴𝑧𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑧𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
1311, 12bitri 278 . 2 (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧𝑥𝐴𝑦𝐴 ((𝑧 ∈ [𝐵]𝑅𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
1410, 13syl6rbbr 293 1 (Rel 𝑅 → (∀𝑥𝐴𝑦𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥𝐴 𝐵𝑅𝑧))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  wal 1536   = wceq 1538  wcel 2112  wral 3109  ∃*wrmo 3112  cin 3883  c0 4246   class class class wbr 5033  Rel wrel 5528  [cec 8274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ec 8278
This theorem is referenced by:  inecmo2  35769  ineccnvmo  35770
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