Theorem inecmo 38550

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 38549	. . 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 3262	. . 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 5108	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐵𝑅𝑧 ↔ 𝐶𝑅𝑧))
6	5	rmo4 3688	. . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝐵𝑅𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐵𝑅𝑧 ∧ 𝐶𝑅𝑧) → 𝑥 = 𝑦))
7		relelec 8682	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑧 ∈ [𝐵]𝑅 ↔ 𝐵𝑅𝑧))
8		relelec 8682	. . . . . . 7 ⊢ (Rel 𝑅 → (𝑧 ∈ [𝐶]𝑅 ↔ 𝐶𝑅𝑧))
9	7, 8	anbi12d 632	. . . . . 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 3200	. . . 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 1921	. 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 206   ∧ wa 395   ∨ wo 847  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃*wrmo 3349   ∩ cin 3900  ∅c0 4285   class class class wbr 5098  Rel wrel 5629  [cec 8633

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rmo 3350  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637

This theorem is referenced by:  inecmo2  38551  ineccnvmo  38552  disjres  39013