Theorem djur 7415

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 7414	. 2 ⊢ (𝐶 ∈ (𝐴 ⊔ 𝐵) ↔ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
2		fvres 5722	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝑥) = (inl‘𝑥))
3	2	eqeq2d 2250	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐶 = (inl‘𝑥)))
4	3	rexbiia 2565	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐶 = (inl‘𝑥))
5		fvres 5722	. . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝑥) = (inr‘𝑥))
6	5	eqeq2d 2250	. . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ 𝐶 = (inr‘𝑥)))
7	6	rexbiia 2565	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ ∃𝑥 ∈ 𝐵 𝐶 = (inr‘𝑥))
8	4, 7	orbi12i 776	. 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)))

This proof depends on syntax axioms:   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   ↾ cres 4778  ‘cfv 5380   ⊔ cdju 7383  inlcinl 7391  inrcinr 7392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-1o 6691  df-dju 7384  df-inl 7393  df-inr 7394

This theorem is used by:  djuss  7416  updjud  7428  omp1eomlem  7440  0ct  7453  ctmlemr  7454  ctssdclemn0  7456  fodjuomnilemdc  7490  exmidfodomrlemeldju  7557