Theorem eldju 6759

Description: Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)

Assertion

Ref	Expression
eldju	⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))

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

Proof of Theorem eldju

Step	Hyp	Ref	Expression
1		djuunr 6758	. . . 4 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)
2	1	eqcomi 2092	. . 3 ⊢ (𝐴 ⊔ 𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
3	2	eleq2i 2154	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4		elun 3141	. . 3 ⊢ (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5		djulf1or 6748	. . . . . 6 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
6		f1ofn 5254	. . . . . 6 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7		fvelrnb 5352	. . . . . 6 ⊢ ((inl ↾ 𝐴) Fn 𝐴 → (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶))
8	5, 6, 7	mp2b 8	. . . . 5 ⊢ (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶)
9		eqcom 2090	. . . . . 6 ⊢ (((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ 𝐶 = ((inl ↾ 𝐴)‘𝑥))
10	9	rexbii 2385	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
11	8, 10	bitri 182	. . . 4 ⊢ (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
12		djurf1or 6749	. . . . . 6 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1𝑜} × 𝐵)
13		f1ofn 5254	. . . . . 6 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1𝑜} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14		fvelrnb 5352	. . . . . 6 ⊢ ((inr ↾ 𝐵) Fn 𝐵 → (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥 ∈ 𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶))
15	12, 13, 14	mp2b 8	. . . . 5 ⊢ (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥 ∈ 𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶)
16		eqcom 2090	. . . . . 6 ⊢ (((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ 𝐶 = ((inr ↾ 𝐵)‘𝑥))
17	16	rexbii 2385	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
18	15, 17	bitri 182	. . . 4 ⊢ (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
19	11, 18	orbi12i 716	. . 3 ⊢ ((𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
20	4, 19	bitri 182	. 2 ⊢ (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
21	3, 20	bitri 182	1 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))

Syntax hints:   ↔ wb 103   ∨ wo 664   = wceq 1289   ∈ wcel 1438  ∃wrex 2360   ∪ cun 2997  ∅c0 3286  {csn 3446   × cxp 4436  ran crn 4439   ↾ cres 4440   Fn wfn 5010  –1-1-onto→wf1o 5014  ‘cfv 5015  1_𝑜c1o 6174   ⊔ cdju 6730  inlcinl 6737  inrcinr 6738

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-dju 6731  df-inl 6739  df-inr 6740

This theorem is referenced by:  exmidfodomrlemreseldju  6826