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Theorem eldju 6921
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 6919 . . . 4 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)
21eqcomi 2121 . . 3 (𝐴𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
32eleq2i 2184 . 2 (𝐶 ∈ (𝐴𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4 elun 3187 . . 3 (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5 djulf1or 6909 . . . . . 6 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
6 f1ofn 5336 . . . . . 6 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7 fvelrnb 5437 . . . . . 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 2119 . . . . . 6 (((inl ↾ 𝐴)‘𝑥) = 𝐶𝐶 = ((inl ↾ 𝐴)‘𝑥))
109rexbii 2419 . . . . 5 (∃𝑥𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ ∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
118, 10bitri 183 . . . 4 (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
12 djurf1or 6910 . . . . . 6 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
13 f1ofn 5336 . . . . . 6 ((inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14 fvelrnb 5437 . . . . . 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 2119 . . . . . 6 (((inr ↾ 𝐵)‘𝑥) = 𝐶𝐶 = ((inr ↾ 𝐵)‘𝑥))
1716rexbii 2419 . . . . 5 (∃𝑥𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
1815, 17bitri 183 . . . 4 (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
1911, 18orbi12i 738 . . 3 ((𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
204, 19bitri 183 . 2 (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
213, 20bitri 183 1 (𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
Colors of variables: wff set class
Syntax hints:  wb 104  wo 682   = wceq 1316  wcel 1465  wrex 2394  cun 3039  c0 3333  {csn 3497   × cxp 4507  ran crn 4510  cres 4511   Fn wfn 5088  1-1-ontowf1o 5092  cfv 5093  1oc1o 6274  cdju 6890  inlcinl 6898  inrcinr 6899
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007  df-1o 6281  df-dju 6891  df-inl 6900  df-inr 6901
This theorem is referenced by:  djur  6922  exmidfodomrlemreseldju  7024
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