Theorem djur 9414

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 9396	. . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2	1	eleq2i 2824	. . 3 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ 𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3		elun 4037	. . 3 ⊢ (𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
4	2, 3	sylbb 222	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
5		xp2nd 7740	. . . 4 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (2nd ‘𝐶) ∈ 𝐴)
6		1st2nd2 7746	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
7		xp1st 7739	. . . . . . 7 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (1st ‘𝐶) ∈ {∅})
8		elsni 4530	. . . . . . 7 ⊢ ((1st ‘𝐶) ∈ {∅} → (1st ‘𝐶) = ∅)
9		opeq1 4756	. . . . . . . 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 235	. . . . 5 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨∅, (2nd ‘𝐶)⟩)
13		fvexd 6683	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (2nd ‘𝐶) ∈ V)
14		opex 5319	. . . . . 6 ⊢ ⟨∅, (2nd ‘𝐶)⟩ ∈ V
15		opeq2 4758	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝐶) → ⟨∅, 𝑦⟩ = ⟨∅, (2nd ‘𝐶)⟩)
16		df-inl 9397	. . . . . . 7 ⊢ inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
17	15, 16	fvmptg 6767	. . . . . 6 ⊢ (((2nd ‘𝐶) ∈ V ∧ ⟨∅, (2nd ‘𝐶)⟩ ∈ V) → (inl‘(2nd ‘𝐶)) = ⟨∅, (2nd ‘𝐶)⟩)
18	13, 14, 17	sylancl 589	. . . . 5 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (inl‘(2nd ‘𝐶)) = ⟨∅, (2nd ‘𝐶)⟩)
19	12, 18	eqtr4d 2776	. . . 4 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = (inl‘(2nd ‘𝐶)))
20		fveq2 6668	. . . . 5 ⊢ (𝑥 = (2nd ‘𝐶) → (inl‘𝑥) = (inl‘(2nd ‘𝐶)))
21	20	rspceeqv 3539	. . . 4 ⊢ (((2nd ‘𝐶) ∈ 𝐴 ∧ 𝐶 = (inl‘(2nd ‘𝐶))) → ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
22	5, 19, 21	syl2anc 587	. . 3 ⊢ (𝐶 ∈ ({∅} × 𝐴) → ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
23		xp2nd 7740	. . . 4 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (2nd ‘𝐶) ∈ 𝐵)
24		1st2nd2 7746	. . . . . 6 ⊢ (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
25		xp1st 7739	. . . . . . 7 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (1st ‘𝐶) ∈ {1o})
26		elsni 4530	. . . . . . 7 ⊢ ((1st ‘𝐶) ∈ {1o} → (1st ‘𝐶) = 1o)
27		opeq1 4756	. . . . . . . 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 235	. . . . 5 ⊢ (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨1o, (2nd ‘𝐶)⟩)
31		fvexd 6683	. . . . . 6 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (2nd ‘𝐶) ∈ V)
32		opex 5319	. . . . . 6 ⊢ ⟨1o, (2nd ‘𝐶)⟩ ∈ V
33		opeq2 4758	. . . . . . 7 ⊢ (𝑧 = (2nd ‘𝐶) → ⟨1o, 𝑧⟩ = ⟨1o, (2nd ‘𝐶)⟩)
34		df-inr 9398	. . . . . . 7 ⊢ inr = (𝑧 ∈ V ↦ ⟨1o, 𝑧⟩)
35	33, 34	fvmptg 6767	. . . . . 6 ⊢ (((2nd ‘𝐶) ∈ V ∧ ⟨1o, (2nd ‘𝐶)⟩ ∈ V) → (inr‘(2nd ‘𝐶)) = ⟨1o, (2nd ‘𝐶)⟩)
36	31, 32, 35	sylancl 589	. . . . 5 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (inr‘(2nd ‘𝐶)) = ⟨1o, (2nd ‘𝐶)⟩)
37	30, 36	eqtr4d 2776	. . . 4 ⊢ (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = (inr‘(2nd ‘𝐶)))
38		fveq2 6668	. . . . 5 ⊢ (𝑥 = (2nd ‘𝐶) → (inr‘𝑥) = (inr‘(2nd ‘𝐶)))
39	38	rspceeqv 3539	. . . 4 ⊢ (((2nd ‘𝐶) ∈ 𝐵 ∧ 𝐶 = (inr‘(2nd ‘𝐶))) → ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
40	23, 37, 39	syl2anc 587	. . 3 ⊢ (𝐶 ∈ ({1o} × 𝐵) → ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
41	22, 40	orim12i 908	. 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 209   ∨ wo 846   = wceq 1542   ∈ wcel 2113  ∃wrex 3054  Vcvv 3397   ∪ cun 3839  ∅c0 4209  {csn 4513  ⟨cop 4519   × cxp 5517  ‘cfv 6333  1^st c1st 7705  2^nd c2nd 7706  1_oc1o 8117   ⊔ cdju 9393  inlcinl 9394  inrcinr 9395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6291  df-fun 6335  df-fv 6341  df-1st 7707  df-2nd 7708  df-dju 9396  df-inl 9397  df-inr 9398

This theorem is referenced by:  djuss  9415  djuun  9421  updjud  9429