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Theorem eldju 6759
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 6758 . . . 4 (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) = (𝐴𝐵)
21eqcomi 2092 . . 3 (𝐴𝐵) = (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵))
32eleq2i 2154 . 2 (𝐶 ∈ (𝐴𝐵) ↔ 𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)))
4 elun 3141 . . 3 (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)))
5 djulf1or 6748 . . . . . 6 (inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴)
6 f1ofn 5254 . . . . . 6 ((inl ↾ 𝐴):𝐴1-1-onto→({∅} × 𝐴) → (inl ↾ 𝐴) Fn 𝐴)
7 fvelrnb 5352 . . . . . 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 2090 . . . . . 6 (((inl ↾ 𝐴)‘𝑥) = 𝐶𝐶 = ((inl ↾ 𝐴)‘𝑥))
109rexbii 2385 . . . . 5 (∃𝑥𝐴 ((inl ↾ 𝐴)‘𝑥) = 𝐶 ↔ ∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
118, 10bitri 182 . . . 4 (𝐶 ∈ ran (inl ↾ 𝐴) ↔ ∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥))
12 djurf1or 6749 . . . . . 6 (inr ↾ 𝐵):𝐵1-1-onto→({1𝑜} × 𝐵)
13 f1ofn 5254 . . . . . 6 ((inr ↾ 𝐵):𝐵1-1-onto→({1𝑜} × 𝐵) → (inr ↾ 𝐵) Fn 𝐵)
14 fvelrnb 5352 . . . . . 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 2090 . . . . . 6 (((inr ↾ 𝐵)‘𝑥) = 𝐶𝐶 = ((inr ↾ 𝐵)‘𝑥))
1716rexbii 2385 . . . . 5 (∃𝑥𝐵 ((inr ↾ 𝐵)‘𝑥) = 𝐶 ↔ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
1815, 17bitri 182 . . . 4 (𝐶 ∈ ran (inr ↾ 𝐵) ↔ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥))
1911, 18orbi12i 716 . . 3 ((𝐶 ∈ ran (inl ↾ 𝐴) ∨ 𝐶 ∈ ran (inr ↾ 𝐵)) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
204, 19bitri 182 . 2 (𝐶 ∈ (ran (inl ↾ 𝐴) ∪ ran (inr ↾ 𝐵)) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
213, 20bitri 182 1 (𝐶 ∈ (𝐴𝐵) ↔ (∃𝑥𝐴 𝐶 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥𝐵 𝐶 = ((inr ↾ 𝐵)‘𝑥)))
Colors of variables: wff set class
Syntax hints:  wb 103  wo 664   = wceq 1289  wcel 1438  wrex 2360  cun 2997  c0 3286  {csn 3446   × cxp 4436  ran crn 4439  cres 4440   Fn wfn 5010  1-1-ontowf1o 5014  cfv 5015  1𝑜c1o 6174  cdju 6730  inlcinl 6737  inrcinr 6738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-dju 6731  df-inl 6739  df-inr 6740
This theorem is referenced by:  exmidfodomrlemreseldju  6826
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