Theorem eldju 7266

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 7264	. . . 4 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)
2	1	eqcomi 2235	. . 3 ⊢ (𝐴 ⊔ 𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
3	2	eleq2i 2298	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4		elun 3348	. . 3 ⊢ (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5		djulf1or 7254	. . . . . 6 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
6		f1ofn 5584	. . . . . 6 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7		fvelrnb 5693	. . . . . 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 2233	. . . . . 6 ⊢ (((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ 𝐶 = ((inl ↾ 𝐴)‘𝑥))
10	9	rexbii 2539	. . . . 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 7255	. . . . . 6 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵)
13		f1ofn 5584	. . . . . 6 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14		fvelrnb 5693	. . . . . 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 2233	. . . . . 6 ⊢ (((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ 𝐶 = ((inr ↾ 𝐵)‘𝑥))
17	16	rexbii 2539	. . . . 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 771	. . 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 715   = wceq 1397   ∈ wcel 2202  ∃wrex 2511   ∪ cun 3198  ∅c0 3494  {csn 3669   × cxp 4723  ran crn 4726   ↾ cres 4727   Fn wfn 5321  –1-1-onto→wf1o 5325  ‘cfv 5326  1_oc1o 6574   ⊔ cdju 7235  inlcinl 7243  inrcinr 7244

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  djur  7267  exmidfodomrlemreseldju  7410