Theorem eldju 7096

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 7094	. . . 4 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)
2	1	eqcomi 2193	. . 3 ⊢ (𝐴 ⊔ 𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
3	2	eleq2i 2256	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4		elun 3291	. . 3 ⊢ (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5		djulf1or 7084	. . . . . 6 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
6		f1ofn 5481	. . . . . 6 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7		fvelrnb 5583	. . . . . 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 2191	. . . . . 6 ⊢ (((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ 𝐶 = ((inl ↾ 𝐴)‘𝑥))
10	9	rexbii 2497	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
11	8, 10	bitri 184	. . . 4 ⊢ (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
12		djurf1or 7085	. . . . . 6 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵)
13		f1ofn 5481	. . . . . 6 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14		fvelrnb 5583	. . . . . 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 2191	. . . . . 6 ⊢ (((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ 𝐶 = ((inr ↾ 𝐵)‘𝑥))
17	16	rexbii 2497	. . . . 5 ⊢ (∃𝑥 ∈ 𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
18	15, 17	bitri 184	. . . 4 ⊢ (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
19	11, 18	orbi12i 765	. . 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 184	. 2 ⊢ (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
21	3, 20	bitri 184	1 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))

Syntax hints:   ↔ wb 105   ∨ wo 709   = wceq 1364   ∈ wcel 2160  ∃wrex 2469   ∪ cun 3142  ∅c0 3437  {csn 3607   × cxp 4642  ran crn 4645   ↾ cres 4646   Fn wfn 5230  –1-1-onto→wf1o 5234  ‘cfv 5235  1_oc1o 6433   ⊔ cdju 7065  inlcinl 7073  inrcinr 7074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6164  df-2nd 6165  df-1o 6440  df-dju 7066  df-inl 7075  df-inr 7076

This theorem is referenced by:  djur  7097  exmidfodomrlemreseldju  7228