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Theorem inecmo2 34434
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.) (Revised by Peter Mazsa, 2-Sep-2021.)
Assertion
Ref Expression
inecmo2 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢𝐴 𝑢𝑅𝑥 ∧ Rel 𝑅))
Distinct variable groups:   𝑢,𝐴,𝑣,𝑥   𝑢,𝑅,𝑣,𝑥

Proof of Theorem inecmo2
StepHypRef Expression
1 id 22 . . 3 (𝑢 = 𝑣𝑢 = 𝑣)
21inecmo 34433 . 2 (Rel 𝑅 → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ ∀𝑥∃*𝑢𝐴 𝑢𝑅𝑥))
32pm5.32ri 567 1 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ Rel 𝑅) ↔ (∀𝑥∃*𝑢𝐴 𝑢𝑅𝑥 ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wo 865  wal 1635   = wceq 1637  wral 3096  ∃*wrmo 3099  cin 3768  c0 4116   class class class wbr 4844  Rel wrel 5316  [cec 7977
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 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
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 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-br 4845  df-opab 4907  df-xp 5317  df-rel 5318  df-cnv 5319  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ec 7981
This theorem is referenced by:  inecmo3  34439
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