Theorem eldju 7372

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 7370	. . . 4 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)
2	1	eqcomi 2238	. . 3 ⊢ (𝐴 ⊔ 𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
3	2	eleq2i 2301	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4		elun 3364	. . 3 ⊢ (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5		djulf1or 7360	. . . . . 6 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
6		f1ofn 5620	. . . . . 6 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7		fvelrnb 5729	. . . . . 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 2236	. . . . . 6 ⊢ (((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ 𝐶 = ((inl ↾ 𝐴)‘𝑥))
10	9	rexbii 2551	. . . . 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 7361	. . . . . 6 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵)
13		f1ofn 5620	. . . . . 6 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14		fvelrnb 5729	. . . . . 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 2236	. . . . . 6 ⊢ (((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ 𝐶 = ((inr ↾ 𝐵)‘𝑥))
17	16	rexbii 2551	. . . . 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 772	. . 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 716   = wceq 1398   ∈ wcel 2205  ∃wrex 2523   ∪ cun 3212  ∅c0 3512  {csn 3694   × cxp 4752  ran crn 4755   ↾ cres 4756   Fn wfn 5352  –1-1-onto→wf1o 5356  ‘cfv 5357  1_oc1o 6653   ⊔ cdju 7341  inlcinl 7349  inrcinr 7350

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-dju 7342  df-inl 7351  df-inr 7352

This theorem is referenced by:  djur  7373  exmidfodomrlemreseldju  7516