Theorem djuun 9879

Description: The disjoint union of two classes is the union of the images of those two classes under right and left injection. (Contributed by Jim Kingdon, 22-Jun-2022.)

Assertion

Ref	Expression
djuun	⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)

Proof of Theorem djuun

Dummy variables 𝑥 𝑦 𝑢 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 4116	. . . 4 ⊢ (𝑥 ∈ ((inl “ 𝐴) ∪ (inr “ 𝐵)) ↔ (𝑥 ∈ (inl “ 𝐴) ∨ 𝑥 ∈ (inr “ 𝐵)))
2		djulf1o 9865	. . . . . . . . . . 11 ⊢ inl:V–1-1-onto→({∅} × V)
3		f1ofn 6801	. . . . . . . . . . 11 ⊢ (inl:V–1-1-onto→({∅} × V) → inl Fn V)
4	2, 3	ax-mp 5	. . . . . . . . . 10 ⊢ inl Fn V
5		ssv 3971	. . . . . . . . . 10 ⊢ 𝐴 ⊆ V
6		fvelimab 6933	. . . . . . . . . 10 ⊢ ((inl Fn V ∧ 𝐴 ⊆ V) → (𝑥 ∈ (inl “ 𝐴) ↔ ∃𝑢 ∈ 𝐴 (inl‘𝑢) = 𝑥))
7	4, 5, 6	mp2an 692	. . . . . . . . 9 ⊢ (𝑥 ∈ (inl “ 𝐴) ↔ ∃𝑢 ∈ 𝐴 (inl‘𝑢) = 𝑥)
8	7	biimpi 216	. . . . . . . 8 ⊢ (𝑥 ∈ (inl “ 𝐴) → ∃𝑢 ∈ 𝐴 (inl‘𝑢) = 𝑥)
9		simprr 772	. . . . . . . . 9 ⊢ ((𝑥 ∈ (inl “ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ (inl‘𝑢) = 𝑥)) → (inl‘𝑢) = 𝑥)
10		vex 3451	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
11		opex 5424	. . . . . . . . . . 11 ⊢ ⟨∅, 𝑢⟩ ∈ V
12		opeq2 4838	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑢 → ⟨∅, 𝑧⟩ = ⟨∅, 𝑢⟩)
13		df-inl 9855	. . . . . . . . . . . 12 ⊢ inl = (𝑧 ∈ V ↦ ⟨∅, 𝑧⟩)
14	12, 13	fvmptg 6966	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ V ∧ ⟨∅, 𝑢⟩ ∈ V) → (inl‘𝑢) = ⟨∅, 𝑢⟩)
15	10, 11, 14	mp2an 692	. . . . . . . . . 10 ⊢ (inl‘𝑢) = ⟨∅, 𝑢⟩
16		0ex 5262	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
17	16	snid 4626	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅}
18		opelxpi 5675	. . . . . . . . . . . 12 ⊢ ((∅ ∈ {∅} ∧ 𝑢 ∈ 𝐴) → ⟨∅, 𝑢⟩ ∈ ({∅} × 𝐴))
19	17, 18	mpan 690	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐴 → ⟨∅, 𝑢⟩ ∈ ({∅} × 𝐴))
20	19	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (inl “ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ (inl‘𝑢) = 𝑥)) → ⟨∅, 𝑢⟩ ∈ ({∅} × 𝐴))
21	15, 20	eqeltrid 2832	. . . . . . . . 9 ⊢ ((𝑥 ∈ (inl “ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ (inl‘𝑢) = 𝑥)) → (inl‘𝑢) ∈ ({∅} × 𝐴))
22	9, 21	eqeltrrd 2829	. . . . . . . 8 ⊢ ((𝑥 ∈ (inl “ 𝐴) ∧ (𝑢 ∈ 𝐴 ∧ (inl‘𝑢) = 𝑥)) → 𝑥 ∈ ({∅} × 𝐴))
23	8, 22	rexlimddv 3140	. . . . . . 7 ⊢ (𝑥 ∈ (inl “ 𝐴) → 𝑥 ∈ ({∅} × 𝐴))
24		elun1 4145	. . . . . . 7 ⊢ (𝑥 ∈ ({∅} × 𝐴) → 𝑥 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
25	23, 24	syl 17	. . . . . 6 ⊢ (𝑥 ∈ (inl “ 𝐴) → 𝑥 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
26		df-dju 9854	. . . . . 6 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
27	25, 26	eleqtrrdi 2839	. . . . 5 ⊢ (𝑥 ∈ (inl “ 𝐴) → 𝑥 ∈ (𝐴 ⊔ 𝐵))
28		djurf1o 9866	. . . . . . . . . . 11 ⊢ inr:V–1-1-onto→({1o} × V)
29		f1ofn 6801	. . . . . . . . . . 11 ⊢ (inr:V–1-1-onto→({1o} × V) → inr Fn V)
30	28, 29	ax-mp 5	. . . . . . . . . 10 ⊢ inr Fn V
31		ssv 3971	. . . . . . . . . 10 ⊢ 𝐵 ⊆ V
32		fvelimab 6933	. . . . . . . . . 10 ⊢ ((inr Fn V ∧ 𝐵 ⊆ V) → (𝑥 ∈ (inr “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (inr‘𝑢) = 𝑥))
33	30, 31, 32	mp2an 692	. . . . . . . . 9 ⊢ (𝑥 ∈ (inr “ 𝐵) ↔ ∃𝑢 ∈ 𝐵 (inr‘𝑢) = 𝑥)
34	33	biimpi 216	. . . . . . . 8 ⊢ (𝑥 ∈ (inr “ 𝐵) → ∃𝑢 ∈ 𝐵 (inr‘𝑢) = 𝑥)
35		simprr 772	. . . . . . . . 9 ⊢ ((𝑥 ∈ (inr “ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ (inr‘𝑢) = 𝑥)) → (inr‘𝑢) = 𝑥)
36		opex 5424	. . . . . . . . . . 11 ⊢ ⟨1o, 𝑢⟩ ∈ V
37		opeq2 4838	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑢 → ⟨1o, 𝑧⟩ = ⟨1o, 𝑢⟩)
38		df-inr 9856	. . . . . . . . . . . 12 ⊢ inr = (𝑧 ∈ V ↦ ⟨1o, 𝑧⟩)
39	37, 38	fvmptg 6966	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ V ∧ ⟨1o, 𝑢⟩ ∈ V) → (inr‘𝑢) = ⟨1o, 𝑢⟩)
40	10, 36, 39	mp2an 692	. . . . . . . . . 10 ⊢ (inr‘𝑢) = ⟨1o, 𝑢⟩
41		1oex 8444	. . . . . . . . . . . . 13 ⊢ 1o ∈ V
42	41	snid 4626	. . . . . . . . . . . 12 ⊢ 1o ∈ {1o}
43		opelxpi 5675	. . . . . . . . . . . 12 ⊢ ((1o ∈ {1o} ∧ 𝑢 ∈ 𝐵) → ⟨1o, 𝑢⟩ ∈ ({1o} × 𝐵))
44	42, 43	mpan 690	. . . . . . . . . . 11 ⊢ (𝑢 ∈ 𝐵 → ⟨1o, 𝑢⟩ ∈ ({1o} × 𝐵))
45	44	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (inr “ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ (inr‘𝑢) = 𝑥)) → ⟨1o, 𝑢⟩ ∈ ({1o} × 𝐵))
46	40, 45	eqeltrid 2832	. . . . . . . . 9 ⊢ ((𝑥 ∈ (inr “ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ (inr‘𝑢) = 𝑥)) → (inr‘𝑢) ∈ ({1o} × 𝐵))
47	35, 46	eqeltrrd 2829	. . . . . . . 8 ⊢ ((𝑥 ∈ (inr “ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ (inr‘𝑢) = 𝑥)) → 𝑥 ∈ ({1o} × 𝐵))
48	34, 47	rexlimddv 3140	. . . . . . 7 ⊢ (𝑥 ∈ (inr “ 𝐵) → 𝑥 ∈ ({1o} × 𝐵))
49		elun2 4146	. . . . . . 7 ⊢ (𝑥 ∈ ({1o} × 𝐵) → 𝑥 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
50	48, 49	syl 17	. . . . . 6 ⊢ (𝑥 ∈ (inr “ 𝐵) → 𝑥 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
51	50, 26	eleqtrrdi 2839	. . . . 5 ⊢ (𝑥 ∈ (inr “ 𝐵) → 𝑥 ∈ (𝐴 ⊔ 𝐵))
52	27, 51	jaoi 857	. . . 4 ⊢ ((𝑥 ∈ (inl “ 𝐴) ∨ 𝑥 ∈ (inr “ 𝐵)) → 𝑥 ∈ (𝐴 ⊔ 𝐵))
53	1, 52	sylbi 217	. . 3 ⊢ (𝑥 ∈ ((inl “ 𝐴) ∪ (inr “ 𝐵)) → 𝑥 ∈ (𝐴 ⊔ 𝐵))
54	53	ssriv 3950	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) ⊆ (𝐴 ⊔ 𝐵)
55		djur 9872	. . . . 5 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → (∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦)))
56		vex 3451	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
57		f1odm 6804	. . . . . . . . . . 11 ⊢ (inl:V–1-1-onto→({∅} × V) → dom inl = V)
58	2, 57	ax-mp 5	. . . . . . . . . 10 ⊢ dom inl = V
59	56, 58	eleqtrri 2827	. . . . . . . . 9 ⊢ 𝑦 ∈ dom inl
60		simpl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑦 ∈ 𝐴)
61	13	funmpt2 6555	. . . . . . . . . 10 ⊢ Fun inl
62		funfvima 7204	. . . . . . . . . 10 ⊢ ((Fun inl ∧ 𝑦 ∈ dom inl) → (𝑦 ∈ 𝐴 → (inl‘𝑦) ∈ (inl “ 𝐴)))
63	61, 62	mpan 690	. . . . . . . . 9 ⊢ (𝑦 ∈ dom inl → (𝑦 ∈ 𝐴 → (inl‘𝑦) ∈ (inl “ 𝐴)))
64	59, 60, 63	mpsyl 68	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → (inl‘𝑦) ∈ (inl “ 𝐴))
65		eleq1 2816	. . . . . . . . 9 ⊢ (𝑥 = (inl‘𝑦) → (𝑥 ∈ (inl “ 𝐴) ↔ (inl‘𝑦) ∈ (inl “ 𝐴)))
66	65	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → (𝑥 ∈ (inl “ 𝐴) ↔ (inl‘𝑦) ∈ (inl “ 𝐴)))
67	64, 66	mpbird 257	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = (inl‘𝑦)) → 𝑥 ∈ (inl “ 𝐴))
68	67	rexlimiva 3126	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ (inl “ 𝐴))
69		f1odm 6804	. . . . . . . . . . 11 ⊢ (inr:V–1-1-onto→({1o} × V) → dom inr = V)
70	28, 69	ax-mp 5	. . . . . . . . . 10 ⊢ dom inr = V
71	56, 70	eleqtrri 2827	. . . . . . . . 9 ⊢ 𝑦 ∈ dom inr
72		simpl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑦 ∈ 𝐵)
73		f1ofun 6802	. . . . . . . . . . 11 ⊢ (inr:V–1-1-onto→({1o} × V) → Fun inr)
74	28, 73	ax-mp 5	. . . . . . . . . 10 ⊢ Fun inr
75		funfvima 7204	. . . . . . . . . 10 ⊢ ((Fun inr ∧ 𝑦 ∈ dom inr) → (𝑦 ∈ 𝐵 → (inr‘𝑦) ∈ (inr “ 𝐵)))
76	74, 75	mpan 690	. . . . . . . . 9 ⊢ (𝑦 ∈ dom inr → (𝑦 ∈ 𝐵 → (inr‘𝑦) ∈ (inr “ 𝐵)))
77	71, 72, 76	mpsyl 68	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → (inr‘𝑦) ∈ (inr “ 𝐵))
78		eleq1 2816	. . . . . . . . 9 ⊢ (𝑥 = (inr‘𝑦) → (𝑥 ∈ (inr “ 𝐵) ↔ (inr‘𝑦) ∈ (inr “ 𝐵)))
79	78	adantl 481	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → (𝑥 ∈ (inr “ 𝐵) ↔ (inr‘𝑦) ∈ (inr “ 𝐵)))
80	77, 79	mpbird 257	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (inr‘𝑦)) → 𝑥 ∈ (inr “ 𝐵))
81	80	rexlimiva 3126	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ (inr “ 𝐵))
82	68, 81	orim12i 908	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦 ∈ 𝐵 𝑥 = (inr‘𝑦)) → (𝑥 ∈ (inl “ 𝐴) ∨ 𝑥 ∈ (inr “ 𝐵)))
83	55, 82	syl 17	. . . 4 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → (𝑥 ∈ (inl “ 𝐴) ∨ 𝑥 ∈ (inr “ 𝐵)))
84	83, 1	sylibr 234	. . 3 ⊢ (𝑥 ∈ (𝐴 ⊔ 𝐵) → 𝑥 ∈ ((inl “ 𝐴) ∪ (inr “ 𝐵)))
85	84	ssriv 3950	. 2 ⊢ (𝐴 ⊔ 𝐵) ⊆ ((inl “ 𝐴) ∪ (inr “ 𝐵))
86	54, 85	eqssi 3963	1 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴 ⊔ 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ∪ cun 3912   ⊆ wss 3914  ∅c0 4296  {csn 4589  ⟨cop 4595   × cxp 5636  dom cdm 5638   “ cima 5641  Fun wfun 6505   Fn wfn 6506  –1-1-onto→wf1o 6510  ‘cfv 6511  1_oc1o 8427   ⊔ cdju 9851  inlcinl 9852  inrcinr 9853

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1st 7968  df-2nd 7969  df-1o 8434  df-dju 9854  df-inl 9855  df-inr 9856

This theorem is referenced by: (None)