Theorem inecmo 37528

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		ineleq 37527	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑧∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
2		ralcom4 3282	. . 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))
3	1, 2	bitri 275	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦))
4		inecmo.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
5	4	breq1d 5158	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵𝑅𝑧 ↔ 𝐶𝑅𝑧))
6	5	rmo4 3726	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦))
7		relelec 8751	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑧 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑧))
8		relelec 8751	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑧 ∈ [𝐶]𝑅 ↔ 𝐶𝑅𝑧))
9	7, 8	anbi12d 630	. . . . . 6 ⊢ (Rel 𝑅 → ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) ↔ (𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧)))
10	9	imbi1d 341	. . . . 5 ⊢ (Rel 𝑅 → (((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦)))
11	10	2ralbidv 3217	. . . 4 ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦)))
12	6, 11	bitr4id 290	. . 3 ⊢ (Rel 𝑅 → (∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
13	12	albidv 1922	. 2 ⊢ (Rel 𝑅 → (∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑧∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝑧 ∈ [𝐵]𝑅 ∧ 𝑧 ∈ [𝐶]𝑅) → 𝑥 = 𝑦)))
14	3, 13	bitr4id 290	1 ⊢ (Rel 𝑅 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 = 𝑦 ∨ ([𝐵]𝑅 ∩ [𝐶]𝑅) = ∅) ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 844  ∀wal 1538   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃*wrmo 3374   ∩ cin 3947  ∅c0 4322   class class class wbr 5148  Rel wrel 5681  [cec 8704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rmo 3375  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8708

This theorem is referenced by:  inecmo2  37529  ineccnvmo  37530  disjres  37918