Theorem inecmo 35613

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 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
inecmo	⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥 ∈ 𝐴 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

Step	Hyp	Ref	Expression
1		relelec 8337	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑧 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑧))
2		relelec 8337	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑧 ∈ [𝐶]𝑅 ↔ 𝐶𝑅𝑧))
3	1, 2	anbi12d 632	. . . . . 6 ⊢ (Rel 𝑅 → ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) ↔ (𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧)))
4	3	imbi1d 344	. . . . 5 ⊢ (Rel 𝑅 → (((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦)))
5	4	2ralbidv 3202	. . . 4 ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦)))
6		inecmo.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
7	6	breq1d 5079	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵𝑅𝑧 ↔ 𝐶𝑅𝑧))
8	7	rmo4 3724	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦))
9	5, 8	syl6rbbr 292	. . 3 ⊢ (Rel 𝑅 → (∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
10	9	albidv 1920	. 2 ⊢ (Rel 𝑅 → (∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑧∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
11		ineleq 35612	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
12		ralcom4 3238	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑧∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
13	11, 12	bitri 277	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
14	10, 13	syl6rbbr 292	1 ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843  ∀wal 1534   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∃*wrmo 3144   ∩ cin 3938  ∅c0 4294   class class class wbr 5069  Rel wrel 5563  [cec 8290

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-xp 5564  df-rel 5565  df-cnv 5566  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-ec 8294

This theorem is referenced by:  inecmo2  35614  ineccnvmo  35615