Theorem djur 7311

Description: A member of a disjoint union can be mapped from one of the classes which produced it. (Contributed by Jim Kingdon, 23-Jun-2022.) Upgrade implication to biconditional and shorten proof. (Revised by BJ, 14-Jul-2023.)

Assertion

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

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

Proof of Theorem djur

Step	Hyp	Ref	Expression
1		eldju 7310	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
2		fvres 5672	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑥) = (inl‘𝑥))
3	2	eqeq2d 2243	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐶 = (inl‘𝑥)))
4	3	rexbiia 2548	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
5		fvres 5672	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑥) = (inr‘𝑥))
6	5	eqeq2d 2243	. . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ 𝐶 = (inr‘𝑥)))
7	6	rexbiia 2548	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
8	4, 7	orbi12i 772	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))
9	1, 8	bitri 184	1 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥)))

Syntax hints:   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2202  ∃wrex 2512   ↾ cres 4733  ‘cfv 5333   ⊔ cdju 7279  inlcinl 7287  inrcinr 7288

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1st 6312  df-2nd 6313  df-1o 6625  df-dju 7280  df-inl 7289  df-inr 7290

This theorem is referenced by:  djuss  7312  updjud  7324  omp1eomlem  7336  0ct  7349  ctmlemr  7350  ctssdclemn0  7352  fodjuomnilemdc  7386  exmidfodomrlemeldju  7453