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Theorem djur 7013
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 7012 . 2 (𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
2 fvres 5492 . . . . 5 (𝑥𝐴 → ((inl ↾ 𝐴)‘𝑥) = (inl‘𝑥))
32eqeq2d 2169 . . . 4 (𝑥𝐴 → (𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐶 = (inl‘𝑥)))
43rexbiia 2472 . . 3 (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ↔ ∃𝑥𝐴 𝐶 = (inl‘𝑥))
5 fvres 5492 . . . . 5 (𝑥𝐵 → ((inr ↾ 𝐵)‘𝑥) = (inr‘𝑥))
65eqeq2d 2169 . . . 4 (𝑥𝐵 → (𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ 𝐶 = (inr‘𝑥)))
76rexbiia 2472 . . 3 (∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥) ↔ ∃𝑥𝐵 𝐶 = (inr‘𝑥))
84, 7orbi12i 754 . 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 698   = wceq 1335  wcel 2128  wrex 2436  cres 4588  cfv 5170  cdju 6981  inlcinl 6989  inrcinr 6990
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-2nd 6089  df-1o 6363  df-dju 6982  df-inl 6991  df-inr 6992
This theorem is referenced by:  djuss  7014  updjud  7026  omp1eomlem  7038  0ct  7051  ctmlemr  7052  ctssdclemn0  7054  fodjuomnilemdc  7087  exmidfodomrlemeldju  7134
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