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Theorem djur 7067
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 (𝐢 ∈ (𝐴 βŠ” 𝐡) ↔ (βˆƒπ‘₯ ∈ 𝐴 𝐢 = (inlβ€˜π‘₯) ∨ βˆƒπ‘₯ ∈ 𝐡 𝐢 = (inrβ€˜π‘₯)))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐢

Proof of Theorem djur
StepHypRef Expression
1 eldju 7066 . 2 (𝐢 ∈ (𝐴 βŠ” 𝐡) ↔ (βˆƒπ‘₯ ∈ 𝐴 𝐢 = ((inl β†Ύ 𝐴)β€˜π‘₯) ∨ βˆƒπ‘₯ ∈ 𝐡 𝐢 = ((inr β†Ύ 𝐡)β€˜π‘₯)))
2 fvres 5539 . . . . 5 (π‘₯ ∈ 𝐴 β†’ ((inl β†Ύ 𝐴)β€˜π‘₯) = (inlβ€˜π‘₯))
32eqeq2d 2189 . . . 4 (π‘₯ ∈ 𝐴 β†’ (𝐢 = ((inl β†Ύ 𝐴)β€˜π‘₯) ↔ 𝐢 = (inlβ€˜π‘₯)))
43rexbiia 2492 . . 3 (βˆƒπ‘₯ ∈ 𝐴 𝐢 = ((inl β†Ύ 𝐴)β€˜π‘₯) ↔ βˆƒπ‘₯ ∈ 𝐴 𝐢 = (inlβ€˜π‘₯))
5 fvres 5539 . . . . 5 (π‘₯ ∈ 𝐡 β†’ ((inr β†Ύ 𝐡)β€˜π‘₯) = (inrβ€˜π‘₯))
65eqeq2d 2189 . . . 4 (π‘₯ ∈ 𝐡 β†’ (𝐢 = ((inr β†Ύ 𝐡)β€˜π‘₯) ↔ 𝐢 = (inrβ€˜π‘₯)))
76rexbiia 2492 . . 3 (βˆƒπ‘₯ ∈ 𝐡 𝐢 = ((inr β†Ύ 𝐡)β€˜π‘₯) ↔ βˆƒπ‘₯ ∈ 𝐡 𝐢 = (inrβ€˜π‘₯))
84, 7orbi12i 764 . 2 ((βˆƒπ‘₯ ∈ 𝐴 𝐢 = ((inl β†Ύ 𝐴)β€˜π‘₯) ∨ βˆƒπ‘₯ ∈ 𝐡 𝐢 = ((inr β†Ύ 𝐡)β€˜π‘₯)) ↔ (βˆƒπ‘₯ ∈ 𝐴 𝐢 = (inlβ€˜π‘₯) ∨ βˆƒπ‘₯ ∈ 𝐡 𝐢 = (inrβ€˜π‘₯)))
91, 8bitri 184 1 (𝐢 ∈ (𝐴 βŠ” 𝐡) ↔ (βˆƒπ‘₯ ∈ 𝐴 𝐢 = (inlβ€˜π‘₯) ∨ βˆƒπ‘₯ ∈ 𝐡 𝐢 = (inrβ€˜π‘₯)))
Colors of variables: wff set class
Syntax hints:   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  βˆƒwrex 2456   β†Ύ cres 4628  β€˜cfv 5216   βŠ” cdju 7035  inlcinl 7043  inrcinr 7044
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 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-1st 6140  df-2nd 6141  df-1o 6416  df-dju 7036  df-inl 7045  df-inr 7046
This theorem is referenced by:  djuss  7068  updjud  7080  omp1eomlem  7092  0ct  7105  ctmlemr  7106  ctssdclemn0  7108  fodjuomnilemdc  7141  exmidfodomrlemeldju  7197
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