Theorem eldju 7069

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 7067	. . . 4 ⊢ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴 ⊔ 𝐵)
2	1	eqcomi 2181	. . 3 ⊢ (𝐴 ⊔ 𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
3	2	eleq2i 2244	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4		elun 3278	. . 3 ⊢ (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5		djulf1or 7057	. . . . . 6 ⊢ (inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴)
6		f1ofn 5464	. . . . . 6 ⊢ ((inl ↾ 𝐴):𝐴–1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7		fvelrnb 5565	. . . . . 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 2179	. . . . . 6 ⊢ (((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ 𝐶 = ((inl ↾ 𝐴)‘𝑥))
10	9	rexbii 2484	. . . . 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 7058	. . . . . 6 ⊢ (inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵)
13		f1ofn 5464	. . . . . 6 ⊢ ((inr ↾ 𝐵):𝐵–1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14		fvelrnb 5565	. . . . . 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 2179	. . . . . 6 ⊢ (((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ 𝐶 = ((inr ↾ 𝐵)‘𝑥))
17	16	rexbii 2484	. . . . 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 764	. . 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 708   = wceq 1353   ∈ wcel 2148  ∃wrex 2456   ∪ cun 3129  ∅c0 3424  {csn 3594   × cxp 4626  ran crn 4629   ↾ cres 4630   Fn wfn 5213  –1-1-onto→wf1o 5217  ‘cfv 5218  1_oc1o 6412   ⊔ cdju 7038  inlcinl 7046  inrcinr 7047

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-dju 7039  df-inl 7048  df-inr 7049

This theorem is referenced by:  djur  7070  exmidfodomrlemreseldju  7201