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Theorem djur 6954
 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 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem djur
StepHypRef Expression
1 eldju 6953 . 2 (𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
2 fvres 5445 . . . . 5 (𝑥𝐴 → ((inl ↾ 𝐴)‘𝑥) = (inl‘𝑥))
32eqeq2d 2151 . . . 4 (𝑥𝐴 → (𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐶 = (inl‘𝑥)))
43rexbiia 2450 . . 3 (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ ∃𝑥𝐴 𝐶 = (inl‘𝑥))
5 fvres 5445 . . . . 5 (𝑥𝐵 → ((inr ↾ 𝐵)‘𝑥) = (inr‘𝑥))
65eqeq2d 2151 . . . 4 (𝑥𝐵 → (𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ 𝐶 = (inr‘𝑥)))
76rexbiia 2450 . . 3 (∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ ∃𝑥𝐵 𝐶 = (inr‘𝑥))
84, 7orbi12i 753 . 2 ((∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)) ↔ (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
91, 8bitri 183 1 (𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   ∨ wo 697   = wceq 1331   ∈ wcel 1480  ∃wrex 2417   ↾ cres 4541  ‘cfv 5123   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-dju 6923  df-inl 6932  df-inr 6933 This theorem is referenced by:  djuss  6955  updjud  6967  omp1eomlem  6979  0ct  6992  ctmlemr  6993  ctssdclemn0  6995  fodjuomnilemdc  7016  exmidfodomrlemeldju  7055
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