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Theorem djur 9677
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.)
Assertion
Ref Expression
djur (𝐶 ∈ (𝐴𝐵) → (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem djur
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dju 9659 . . . 4 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2830 . . 3 (𝐶 ∈ (𝐴𝐵) ↔ 𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 4083 . . 3 (𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
42, 3sylbb 218 . 2 (𝐶 ∈ (𝐴𝐵) → (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
5 xp2nd 7864 . . . 4 (𝐶 ∈ ({∅} × 𝐴) → (2nd𝐶) ∈ 𝐴)
6 1st2nd2 7870 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
7 xp1st 7863 . . . . . . 7 (𝐶 ∈ ({∅} × 𝐴) → (1st𝐶) ∈ {∅})
8 elsni 4578 . . . . . . 7 ((1st𝐶) ∈ {∅} → (1st𝐶) = ∅)
9 opeq1 4804 . . . . . . . 8 ((1st𝐶) = ∅ → ⟨(1st𝐶), (2nd𝐶)⟩ = ⟨∅, (2nd𝐶)⟩)
109eqeq2d 2749 . . . . . . 7 ((1st𝐶) = ∅ → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd𝐶)⟩))
117, 8, 103syl 18 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd𝐶)⟩))
126, 11mpbid 231 . . . . 5 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨∅, (2nd𝐶)⟩)
13 fvexd 6789 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → (2nd𝐶) ∈ V)
14 opex 5379 . . . . . 6 ⟨∅, (2nd𝐶)⟩ ∈ V
15 opeq2 4805 . . . . . . 7 (𝑦 = (2nd𝐶) → ⟨∅, 𝑦⟩ = ⟨∅, (2nd𝐶)⟩)
16 df-inl 9660 . . . . . . 7 inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
1715, 16fvmptg 6873 . . . . . 6 (((2nd𝐶) ∈ V ∧ ⟨∅, (2nd𝐶)⟩ ∈ V) → (inl‘(2nd𝐶)) = ⟨∅, (2nd𝐶)⟩)
1813, 14, 17sylancl 586 . . . . 5 (𝐶 ∈ ({∅} × 𝐴) → (inl‘(2nd𝐶)) = ⟨∅, (2nd𝐶)⟩)
1912, 18eqtr4d 2781 . . . 4 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = (inl‘(2nd𝐶)))
20 fveq2 6774 . . . . 5 (𝑥 = (2nd𝐶) → (inl‘𝑥) = (inl‘(2nd𝐶)))
2120rspceeqv 3575 . . . 4 (((2nd𝐶) ∈ 𝐴𝐶 = (inl‘(2nd𝐶))) → ∃𝑥𝐴 𝐶 = (inl‘𝑥))
225, 19, 21syl2anc 584 . . 3 (𝐶 ∈ ({∅} × 𝐴) → ∃𝑥𝐴 𝐶 = (inl‘𝑥))
23 xp2nd 7864 . . . 4 (𝐶 ∈ ({1o} × 𝐵) → (2nd𝐶) ∈ 𝐵)
24 1st2nd2 7870 . . . . . 6 (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
25 xp1st 7863 . . . . . . 7 (𝐶 ∈ ({1o} × 𝐵) → (1st𝐶) ∈ {1o})
26 elsni 4578 . . . . . . 7 ((1st𝐶) ∈ {1o} → (1st𝐶) = 1o)
27 opeq1 4804 . . . . . . . 8 ((1st𝐶) = 1o → ⟨(1st𝐶), (2nd𝐶)⟩ = ⟨1o, (2nd𝐶)⟩)
2827eqeq2d 2749 . . . . . . 7 ((1st𝐶) = 1o → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨1o, (2nd𝐶)⟩))
2925, 26, 283syl 18 . . . . . 6 (𝐶 ∈ ({1o} × 𝐵) → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨1o, (2nd𝐶)⟩))
3024, 29mpbid 231 . . . . 5 (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨1o, (2nd𝐶)⟩)
31 fvexd 6789 . . . . . 6 (𝐶 ∈ ({1o} × 𝐵) → (2nd𝐶) ∈ V)
32 opex 5379 . . . . . 6 ⟨1o, (2nd𝐶)⟩ ∈ V
33 opeq2 4805 . . . . . . 7 (𝑧 = (2nd𝐶) → ⟨1o, 𝑧⟩ = ⟨1o, (2nd𝐶)⟩)
34 df-inr 9661 . . . . . . 7 inr = (𝑧 ∈ V ↦ ⟨1o, 𝑧⟩)
3533, 34fvmptg 6873 . . . . . 6 (((2nd𝐶) ∈ V ∧ ⟨1o, (2nd𝐶)⟩ ∈ V) → (inr‘(2nd𝐶)) = ⟨1o, (2nd𝐶)⟩)
3631, 32, 35sylancl 586 . . . . 5 (𝐶 ∈ ({1o} × 𝐵) → (inr‘(2nd𝐶)) = ⟨1o, (2nd𝐶)⟩)
3730, 36eqtr4d 2781 . . . 4 (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = (inr‘(2nd𝐶)))
38 fveq2 6774 . . . . 5 (𝑥 = (2nd𝐶) → (inr‘𝑥) = (inr‘(2nd𝐶)))
3938rspceeqv 3575 . . . 4 (((2nd𝐶) ∈ 𝐵𝐶 = (inr‘(2nd𝐶))) → ∃𝑥𝐵 𝐶 = (inr‘𝑥))
4023, 37, 39syl2anc 584 . . 3 (𝐶 ∈ ({1o} × 𝐵) → ∃𝑥𝐵 𝐶 = (inr‘𝑥))
4122, 40orim12i 906 . 2 ((𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)) → (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
424, 41syl 17 1 (𝐶 ∈ (𝐴𝐵) → (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 844   = wceq 1539  wcel 2106  wrex 3065  Vcvv 3432  cun 3885  c0 4256  {csn 4561  cop 4567   × cxp 5587  cfv 6433  1st c1st 7829  2nd c2nd 7830  1oc1o 8290  cdju 9656  inlcinl 9657  inrcinr 9658
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831  df-2nd 7832  df-dju 9659  df-inl 9660  df-inr 9661
This theorem is referenced by:  djuss  9678  djuun  9684  updjud  9692
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