Theorem djur 8945

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 8928	. . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1𝑜} × 𝐵))
2	1	eleq2i 2842	. . 3 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ 𝐶 ∈ (({∅} × 𝐴) ∪ ({1𝑜} × 𝐵)))
3		elun 3904	. . 3 ⊢ (𝐶 ∈ (({∅} × 𝐴) ∪ ({1𝑜} × 𝐵)) ↔ (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1𝑜} × 𝐵)))
4	2, 3	sylbb 209	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1𝑜} × 𝐵)))
5		xp2nd 7348	. . . 4 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (2nd ‘𝐶) ∈ 𝐴)
6		1st2nd2 7354	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
7		xp1st 7347	. . . . . . 7 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (1st ‘𝐶) ∈ {∅})
8		elsni 4333	. . . . . . 7 ⊢ ((1st ‘𝐶) ∈ {∅} → (1st ‘𝐶) = ∅)
9		opeq1 4539	. . . . . . . 8 ⊢ ((1st ‘𝐶) = ∅ → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ = ⟨∅, (2nd ‘𝐶)⟩)
10	9	eqeq2d 2781	. . . . . . 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 222	. . . . 5 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨∅, (2nd ‘𝐶)⟩)
13		fvexd 6344	. . . . . 6 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (2nd ‘𝐶) ∈ V)
14		opex 5060	. . . . . 6 ⊢ ⟨∅, (2nd ‘𝐶)⟩ ∈ V
15		opeq2 4540	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝐶) → ⟨∅, 𝑦⟩ = ⟨∅, (2nd ‘𝐶)⟩)
16		df-inl 8929	. . . . . . 7 ⊢ inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
17	15, 16	fvmptg 6422	. . . . . 6 ⊢ (((2nd ‘𝐶) ∈ V ∧ ⟨∅, (2nd ‘𝐶)⟩ ∈ V) → (inl‘(2nd ‘𝐶)) = ⟨∅, (2nd ‘𝐶)⟩)
18	13, 14, 17	sylancl 574	. . . . 5 ⊢ (𝐶 ∈ ({∅} × 𝐴) → (inl‘(2nd ‘𝐶)) = ⟨∅, (2nd ‘𝐶)⟩)
19	12, 18	eqtr4d 2808	. . . 4 ⊢ (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = (inl‘(2nd ‘𝐶)))
20		fveq2 6332	. . . . . 6 ⊢ (𝑥 = (2nd ‘𝐶) → (inl‘𝑥) = (inl‘(2nd ‘𝐶)))
21	20	eqeq2d 2781	. . . . 5 ⊢ (𝑥 = (2nd ‘𝐶) → (𝐶 = (inl‘𝑥) ↔ 𝐶 = (inl‘(2nd ‘𝐶))))
22	21	rspcev 3460	. . . 4 ⊢ (((2nd ‘𝐶) ∈ 𝐴 ∧ 𝐶 = (inl‘(2nd ‘𝐶))) → ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
23	5, 19, 22	syl2anc 573	. . 3 ⊢ (𝐶 ∈ ({∅} × 𝐴) → ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
24		xp2nd 7348	. . . 4 ⊢ (𝐶 ∈ ({1𝑜} × 𝐵) → (2nd ‘𝐶) ∈ 𝐵)
25		1st2nd2 7354	. . . . . 6 ⊢ (𝐶 ∈ ({1𝑜} × 𝐵) → 𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩)
26		xp1st 7347	. . . . . . 7 ⊢ (𝐶 ∈ ({1𝑜} × 𝐵) → (1st ‘𝐶) ∈ {1𝑜})
27		elsni 4333	. . . . . . 7 ⊢ ((1st ‘𝐶) ∈ {1𝑜} → (1st ‘𝐶) = 1𝑜)
28		opeq1 4539	. . . . . . . 8 ⊢ ((1st ‘𝐶) = 1𝑜 → ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ = ⟨1𝑜, (2nd ‘𝐶)⟩)
29	28	eqeq2d 2781	. . . . . . 7 ⊢ ((1st ‘𝐶) = 1𝑜 → (𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ↔ 𝐶 = ⟨1𝑜, (2nd ‘𝐶)⟩))
30	26, 27, 29	3syl 18	. . . . . 6 ⊢ (𝐶 ∈ ({1𝑜} × 𝐵) → (𝐶 = ⟨(1st ‘𝐶), (2nd ‘𝐶)⟩ ↔ 𝐶 = ⟨1𝑜, (2nd ‘𝐶)⟩))
31	25, 30	mpbid 222	. . . . 5 ⊢ (𝐶 ∈ ({1𝑜} × 𝐵) → 𝐶 = ⟨1𝑜, (2nd ‘𝐶)⟩)
32		fvexd 6344	. . . . . 6 ⊢ (𝐶 ∈ ({1𝑜} × 𝐵) → (2nd ‘𝐶) ∈ V)
33		opex 5060	. . . . . 6 ⊢ ⟨1𝑜, (2nd ‘𝐶)⟩ ∈ V
34		opeq2 4540	. . . . . . 7 ⊢ (𝑧 = (2nd ‘𝐶) → ⟨1𝑜, 𝑧⟩ = ⟨1𝑜, (2nd ‘𝐶)⟩)
35		df-inr 8930	. . . . . . 7 ⊢ inr = (𝑧 ∈ V ↦ ⟨1𝑜, 𝑧⟩)
36	34, 35	fvmptg 6422	. . . . . 6 ⊢ (((2nd ‘𝐶) ∈ V ∧ ⟨1𝑜, (2nd ‘𝐶)⟩ ∈ V) → (inr‘(2nd ‘𝐶)) = ⟨1𝑜, (2nd ‘𝐶)⟩)
37	32, 33, 36	sylancl 574	. . . . 5 ⊢ (𝐶 ∈ ({1𝑜} × 𝐵) → (inr‘(2nd ‘𝐶)) = ⟨1𝑜, (2nd ‘𝐶)⟩)
38	31, 37	eqtr4d 2808	. . . 4 ⊢ (𝐶 ∈ ({1𝑜} × 𝐵) → 𝐶 = (inr‘(2nd ‘𝐶)))
39		fveq2 6332	. . . . . 6 ⊢ (𝑥 = (2nd ‘𝐶) → (inr‘𝑥) = (inr‘(2nd ‘𝐶)))
40	39	eqeq2d 2781	. . . . 5 ⊢ (𝑥 = (2nd ‘𝐶) → (𝐶 = (inr‘𝑥) ↔ 𝐶 = (inr‘(2nd ‘𝐶))))
41	40	rspcev 3460	. . . 4 ⊢ (((2nd ‘𝐶) ∈ 𝐵 ∧ 𝐶 = (inr‘(2nd ‘𝐶))) → ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
42	24, 38, 41	syl2anc 573	. . 3 ⊢ (𝐶 ∈ ({1𝑜} × 𝐵) → ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
43	23, 42	orim12i 892	. 2 ⊢ ((𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1𝑜} × 𝐵)) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
44	4, 43	syl 17	1 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) → (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 834   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  Vcvv 3351   ∪ cun 3721  ∅c0 4063  {csn 4316  ⟨cop 4322   × cxp 5247  ‘cfv 6031  1^st c1st 7313  2^nd c2nd 7314  1_𝑜c1o 7706   ⊔ cdju 8925  inlcinl 8926  inrcinr 8927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7315  df-2nd 7316  df-dju 8928  df-inl 8929  df-inr 8930

This theorem is referenced by:  djuss  8946  djuun  8952  updjud  8960