Theorem djur 10037

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 10019	. . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
2	1	eleq2i 2856	. . 3 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ 𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3		elun 4155	. . 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 8068	. . . 4 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (2nd ‘𝐶) ∈ 𝐴)
6		1st2nd2 8074	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
7		xp1st 8067	. . . . . . 7 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (1st ‘𝐶) ∈ {∅})
8		elsni 4601	. . . . . . 7 ⊢ ((1st ‘𝐶) ∈ {∅} → (1st ‘𝐶) = ∅)
9		opeq1 4845	. . . . . . . 8 ⊢ ((1st ‘𝐶) = ∅ → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ = ⟨∅, (2nd ‘𝐶)⟩)
10	9	eqeq2d 2774	. . . . . . 7 ⊢ ((1st ‘𝐶) = ∅ → (𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd ‘𝐶)⟩))
11	7, 8, 10	3syl 19	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd ‘𝐶)⟩))
12	6, 11	mpbid 235	. . . . 5 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨∅, (2nd ‘𝐶)⟩)
13		fvexd 6931	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (2nd ‘𝐶) ∈ V)
14		opex 5439	. . . . . 6 ⊢ ⟨∅, (2nd ‘𝐶)⟩ ∈ V
15		opeq2 4846	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝐶) → ⟨∅, 𝑦⟩ = ⟨∅, (2nd ‘𝐶)⟩)
16		df-inl 10020	. . . . . . 7 ⊢ inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
17	15, 16	fvmptg 7022	. . . . . 6 ⊢ (((2nd ‘𝐶) ∈ V ∧ ⟨∅, (2nd ‘𝐶)⟩ ∈ V) → (inl‘(2nd ‘𝐶)) = ⟨∅, (2nd ‘𝐶)⟩)
18	13, 14, 17	sylancl 598	. . . . 5 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (inl‘(2nd ‘𝐶)) = ⟨∅, (2nd ‘𝐶)⟩)
19	12, 18	eqtr4d 2801	. . . 4 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = (inl‘(2nd ‘𝐶)))
20		fveq2 6916	. . . . 5 ⊢ (𝑥 = (2nd ‘𝐶) → (inl‘𝑥) = (inl‘(2nd ‘𝐶)))
21	20	rspceeqv 3602	. . . 4 ⊢ (((2nd ‘𝐶) ∈ 𝐴 ∧ 𝐶 = (inl‘(2nd ‘𝐶))) → ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
22	5, 19, 21	syl2anc 596	. . 3 ⊢ (𝐶 ∈ ({∅} × 𝐴) → ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
23		xp2nd 8068	. . . 4 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (2nd ‘𝐶) ∈ 𝐵)
24		1st2nd2 8074	. . . . . 6 ⊢ (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
25		xp1st 8067	. . . . . . 7 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (1st ‘𝐶) ∈ {1o})
26		elsni 4601	. . . . . . 7 ⊢ ((1st ‘𝐶) ∈ {1o} → (1st ‘𝐶) = 1o)
27		opeq1 4845	. . . . . . . 8 ⊢ ((1st ‘𝐶) = 1o → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ = ⟨1o, (2nd ‘𝐶)⟩)
28	27	eqeq2d 2774	. . . . . . 7 ⊢ ((1st ‘𝐶) = 1o → (𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ↔ 𝐶 = ⟨1o, (2nd ‘𝐶)⟩))
29	25, 26, 28	3syl 19	. . . . . 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 6931	. . . . . 6 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (2nd ‘𝐶) ∈ V)
32		opex 5439	. . . . . 6 ⊢ ⟨1o, (2nd ‘𝐶)⟩ ∈ V
33		opeq2 4846	. . . . . . 7 ⊢ (𝑧 = (2nd ‘𝐶) → ⟨1o, 𝑧⟩ = ⟨1o, (2nd ‘𝐶)⟩)
34		df-inr 10021	. . . . . . 7 ⊢ inr = (𝑧 ∈ V ↦ ⟨1o, 𝑧⟩)
35	33, 34	fvmptg 7022	. . . . . 6 ⊢ (((2nd ‘𝐶) ∈ V ∧ ⟨1o, (2nd ‘𝐶)⟩ ∈ V) → (inr‘(2nd ‘𝐶)) = ⟨1o, (2nd ‘𝐶)⟩)
36	31, 32, 35	sylancl 598	. . . . 5 ⊢ (𝐶 ∈ ({1o} × 𝐵) → (inr‘(2nd ‘𝐶)) = ⟨1o, (2nd ‘𝐶)⟩)
37	30, 36	eqtr4d 2801	. . . 4 ⊢ (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = (inr‘(2nd ‘𝐶)))
38		fveq2 6916	. . . . 5 ⊢ (𝑥 = (2nd ‘𝐶) → (inr‘𝑥) = (inr‘(2nd ‘𝐶)))
39	38	rspceeqv 3602	. . . 4 ⊢ (((2nd ‘𝐶) ∈ 𝐵 ∧ 𝐶 = (inr‘(2nd ‘𝐶))) → ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
40	23, 37, 39	syl2anc 596	. . 3 ⊢ (𝐶 ∈ ({1o} × 𝐵) → ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
41	22, 40	orim12i 922	. 2 ⊢ ((𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
42	4, 41	syl 18	1 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4840   × cxp 5656  ‘cfv 6566  1^st c1st 8033  2^nd c2nd 8034  1_oc1o 8507   ⊔ cdju 10016  inlcinl 10017  inrcinr 10018

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fv 6574  df-1st 8035  df-2nd 8036  df-dju 10019  df-inl 10020  df-inr 10021

This theorem is used by:  djuss  10038  djuun  10044  updjud  10052