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Theorem eldju 6919
 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 6917 . . . 4 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)
21eqcomi 2119 . . 3 (𝐴𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
32eleq2i 2182 . 2 (𝐶 ∈ (𝐴𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4 elun 3185 . . 3 (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5 djulf1or 6907 . . . . . 6 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
6 f1ofn 5334 . . . . . 6 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7 fvelrnb 5435 . . . . . 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 2117 . . . . . 6 (((inl ↾ 𝐴)‘𝑥) = 𝐶𝐶 = ((inl ↾ 𝐴)‘𝑥))
109rexbii 2417 . . . . 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 6908 . . . . . 6 (inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵)
13 f1ofn 5334 . . . . . 6 ((inr ↾ 𝐵):𝐵1-1-onto→({1o} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14 fvelrnb 5435 . . . . . 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 2117 . . . . . 6 (((inr ↾ 𝐵)‘𝑥) = 𝐶𝐶 = ((inr ↾ 𝐵)‘𝑥))
1716rexbii 2417 . . . . 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 736 . . 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 680   = wceq 1314   ∈ wcel 1463  ∃wrex 2392   ∪ cun 3037  ∅c0 3331  {csn 3495   × cxp 4505  ran crn 4508   ↾ cres 4509   Fn wfn 5086  –1-1-onto→wf1o 5090  ‘cfv 5091  1oc1o 6272   ⊔ cdju 6888  inlcinl 6896  inrcinr 6897 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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005  df-1o 6279  df-dju 6889  df-inl 6898  df-inr 6899 This theorem is referenced by:  djur  6920  exmidfodomrlemreseldju  7020
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