Theorem djur 9677

Description: A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.)

Assertion

Ref	Expression
djur	⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem djur

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

Step	Hyp	Ref	Expression
1		df-dju 9659	. . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2	1	eleq2i 2830	. . 3 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ 𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3		elun 4083	. . 3 ⊢ (𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
4	2, 3	sylbb 218	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
5		xp2nd 7864	. . . 4 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (2nd ‘𝐶) ∈ 𝐴)
6		1st2nd2 7870	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
7		xp1st 7863	. . . . . . 7 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (1st ‘𝐶) ∈ {∅})
8		elsni 4578	. . . . . . 7 ⊢ ((1st ‘𝐶) ∈ {∅} → (1st ‘𝐶) = ∅)
9		opeq1 4804	. . . . . . . 8 ⊢ ((1st ‘𝐶) = ∅ → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ = ⟨∅, (2nd ‘𝐶)⟩)
10	9	eqeq2d 2749	. . . . . . 7 ⊢ ((1st ‘𝐶) = ∅ → (𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd ‘𝐶)⟩))
11	7, 8, 10	3syl 18	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd ‘𝐶)⟩))
12	6, 11	mpbid 231	. . . . 5 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨∅, (2nd ‘𝐶)⟩)
13		fvexd 6789	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (2nd ‘𝐶) ∈ V)
14		opex 5379	. . . . . 6 ⊢ ⟨∅, (2nd ‘𝐶)⟩ ∈ V
15		opeq2 4805	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝐶) → ⟨∅, 𝑦⟩ = ⟨∅, (2nd ‘𝐶)⟩)
16		df-inl 9660	. . . . . . 7 ⊢ inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
17	15, 16	fvmptg 6873	. . . . . 6 ⊢ (((2nd ‘𝐶) ∈ V ∧ ⟨∅, (2nd ‘𝐶)⟩ ∈ V) → (inl‘(2nd ‘𝐶)) = ⟨∅, (2nd ‘𝐶)⟩)
18	13, 14, 17	sylancl 586	. . . . 5 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (inl‘(2nd ‘𝐶)) = ⟨∅, (2nd ‘𝐶)⟩)
19	12, 18	eqtr4d 2781	. . . 4 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = (inl‘(2nd ‘𝐶)))
20		fveq2 6774	. . . . 5 ⊢ (𝑥 = (2nd ‘𝐶) → (inl‘𝑥) = (inl‘(2nd ‘𝐶)))
21	20	rspceeqv 3575	. . . 4 ⊢ (((2nd ‘𝐶) ∈ 𝐴 ∧ 𝐶 = (inl‘(2nd ‘𝐶))) → ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
22	5, 19, 21	syl2anc 584	. . 3 ⊢ (𝐶 ∈ ({∅} × 𝐴) → ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
23		xp2nd 7864	. . . 4 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (2nd ‘𝐶) ∈ 𝐵)
24		1st2nd2 7870	. . . . . 6 ⊢ (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
25		xp1st 7863	. . . . . . 7 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (1st ‘𝐶) ∈ {1o})
26		elsni 4578	. . . . . . 7 ⊢ ((1st ‘𝐶) ∈ {1o} → (1st ‘𝐶) = 1o)
27		opeq1 4804	. . . . . . . 8 ⊢ ((1st ‘𝐶) = 1o → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ = ⟨1o, (2nd ‘𝐶)⟩)
28	27	eqeq2d 2749	. . . . . . 7 ⊢ ((1st ‘𝐶) = 1o → (𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ↔ 𝐶 = ⟨1o, (2nd ‘𝐶)⟩))
29	25, 26, 28	3syl 18	. . . . . 6 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ↔ 𝐶 = ⟨1o, (2nd ‘𝐶)⟩))
30	24, 29	mpbid 231	. . . . 5 ⊢ (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨1o, (2nd ‘𝐶)⟩)
31		fvexd 6789	. . . . . 6 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (2nd ‘𝐶) ∈ V)
32		opex 5379	. . . . . 6 ⊢ ⟨1o, (2nd ‘𝐶)⟩ ∈ V
33		opeq2 4805	. . . . . . 7 ⊢ (𝑧 = (2nd ‘𝐶) → ⟨1o, 𝑧⟩ = ⟨1o, (2nd ‘𝐶)⟩)
34		df-inr 9661	. . . . . . 7 ⊢ inr = (𝑧 ∈ V ↦ ⟨1o, 𝑧⟩)
35	33, 34	fvmptg 6873	. . . . . 6 ⊢ (((2nd ‘𝐶) ∈ V ∧ ⟨1o, (2nd ‘𝐶)⟩ ∈ V) → (inr‘(2nd ‘𝐶)) = ⟨1o, (2nd ‘𝐶)⟩)
36	31, 32, 35	sylancl 586	. . . . 5 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (inr‘(2nd ‘𝐶)) = ⟨1o, (2nd ‘𝐶)⟩)
37	30, 36	eqtr4d 2781	. . . 4 ⊢ (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = (inr‘(2nd ‘𝐶)))
38		fveq2 6774	. . . . 5 ⊢ (𝑥 = (2nd ‘𝐶) → (inr‘𝑥) = (inr‘(2nd ‘𝐶)))
39	38	rspceeqv 3575	. . . 4 ⊢ (((2nd ‘𝐶) ∈ 𝐵 ∧ 𝐶 = (inr‘(2nd ‘𝐶))) → ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
40	23, 37, 39	syl2anc 584	. . 3 ⊢ (𝐶 ∈ ({1o} × 𝐵) → ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
41	22, 40	orim12i 906	. 2 ⊢ ((𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
42	4, 41	syl 17	1 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  Vcvv 3432   ∪ cun 3885  ∅c0 4256  {csn 4561  ⟨cop 4567   × cxp 5587  ‘cfv 6433  1^st c1st 7829  2^nd c2nd 7830  1_oc1o 8290   ⊔ cdju 9656  inlcinl 9657  inrcinr 9658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831  df-2nd 7832  df-dju 9659  df-inl 9660  df-inr 9661

This theorem is referenced by:  djuss  9678  djuun  9684  updjud  9692