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Theorem eldju 7096
Description: Element of a disjoint union. (Contributed by BJ and Jim Kingdon, 23-Jun-2022.)
Assertion
Ref Expression
eldju (𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem eldju
StepHypRef Expression
1 djuunr 7094 . . . 4 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)
21eqcomi 2193 . . 3 (𝐴𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
32eleq2i 2256 . 2 (𝐶 ∈ (𝐴𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4 elun 3291 . . 3 (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5 djulf1or 7084 . . . . . 6 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
6 f1ofn 5481 . . . . . 6 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7 fvelrnb 5583 . . . . . 6 ((inl ↾ 𝐴) Fn 𝐴 → (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶))
85, 6, 7mp2b 8 . . . . 5 (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶)
9 eqcom 2191 . . . . . 6 (((inl ↾ 𝐴)‘𝑥) = 𝐶𝐶 = ((inl ↾ 𝐴)‘𝑥))
109rexbii 2497 . . . . 5 (∃𝑥𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ ∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
118, 10bitri 184 . . . 4 (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
12 djurf1or 7085 . . . . . 6 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
13 f1ofn 5481 . . . . . 6 ((inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14 fvelrnb 5583 . . . . . 6 ((inr ↾ 𝐵) Fn 𝐵 → (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶))
1512, 13, 14mp2b 8 . . . . 5 (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶)
16 eqcom 2191 . . . . . 6 (((inr ↾ 𝐵)‘𝑥) = 𝐶𝐶 = ((inr ↾ 𝐵)‘𝑥))
1716rexbii 2497 . . . . 5 (∃𝑥𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
1815, 17bitri 184 . . . 4 (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
1911, 18orbi12i 765 . . 3 ((𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
204, 19bitri 184 . 2 (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
213, 20bitri 184 1 (𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
Colors of variables: wff set class
Syntax hints:  wb 105  wo 709   = wceq 1364  wcel 2160  wrex 2469  cun 3142  c0 3437  {csn 3607   × cxp 4642  ran crn 4645  cres 4646   Fn wfn 5230  1-1-ontowf1o 5234  cfv 5235  1oc1o 6433  cdju 7065  inlcinl 7073  inrcinr 7074
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6164  df-2nd 6165  df-1o 6440  df-dju 7066  df-inl 7075  df-inr 7076
This theorem is referenced by:  djur  7097  exmidfodomrlemreseldju  7228
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