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Theorem djur 8945
 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 8928 . . . 4 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1𝑜} × 𝐵))
21eleq2i 2842 . . 3 (𝐶 ∈ (𝐴𝐵) ↔ 𝐶 ∈ (({∅} × 𝐴) ∪ ({1𝑜} × 𝐵)))
3 elun 3904 . . 3 (𝐶 ∈ (({∅} × 𝐴) ∪ ({1𝑜} × 𝐵)) ↔ (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1𝑜} × 𝐵)))
42, 3sylbb 209 . 2 (𝐶 ∈ (𝐴𝐵) → (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1𝑜} × 𝐵)))
5 xp2nd 7348 . . . 4 (𝐶 ∈ ({∅} × 𝐴) → (2nd𝐶) ∈ 𝐴)
6 1st2nd2 7354 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
7 xp1st 7347 . . . . . . 7 (𝐶 ∈ ({∅} × 𝐴) → (1st𝐶) ∈ {∅})
8 elsni 4333 . . . . . . 7 ((1st𝐶) ∈ {∅} → (1st𝐶) = ∅)
9 opeq1 4539 . . . . . . . 8 ((1st𝐶) = ∅ → ⟨(1st𝐶), (2nd𝐶)⟩ = ⟨∅, (2nd𝐶)⟩)
109eqeq2d 2781 . . . . . . 7 ((1st𝐶) = ∅ → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd𝐶)⟩))
117, 8, 103syl 18 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd𝐶)⟩))
126, 11mpbid 222 . . . . 5 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨∅, (2nd𝐶)⟩)
13 fvexd 6344 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → (2nd𝐶) ∈ V)
14 opex 5060 . . . . . 6 ⟨∅, (2nd𝐶)⟩ ∈ V
15 opeq2 4540 . . . . . . 7 (𝑦 = (2nd𝐶) → ⟨∅, 𝑦⟩ = ⟨∅, (2nd𝐶)⟩)
16 df-inl 8929 . . . . . . 7 inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
1715, 16fvmptg 6422 . . . . . 6 (((2nd𝐶) ∈ V ∧ ⟨∅, (2nd𝐶)⟩ ∈ V) → (inl‘(2nd𝐶)) = ⟨∅, (2nd𝐶)⟩)
1813, 14, 17sylancl 574 . . . . 5 (𝐶 ∈ ({∅} × 𝐴) → (inl‘(2nd𝐶)) = ⟨∅, (2nd𝐶)⟩)
1912, 18eqtr4d 2808 . . . 4 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = (inl‘(2nd𝐶)))
20 fveq2 6332 . . . . . 6 (𝑥 = (2nd𝐶) → (inl‘𝑥) = (inl‘(2nd𝐶)))
2120eqeq2d 2781 . . . . 5 (𝑥 = (2nd𝐶) → (𝐶 = (inl‘𝑥) ↔ 𝐶 = (inl‘(2nd𝐶))))
2221rspcev 3460 . . . 4 (((2nd𝐶) ∈ 𝐴𝐶 = (inl‘(2nd𝐶))) → ∃𝑥𝐴 𝐶 = (inl‘𝑥))
235, 19, 22syl2anc 573 . . 3 (𝐶 ∈ ({∅} × 𝐴) → ∃𝑥𝐴 𝐶 = (inl‘𝑥))
24 xp2nd 7348 . . . 4 (𝐶 ∈ ({1𝑜} × 𝐵) → (2nd𝐶) ∈ 𝐵)
25 1st2nd2 7354 . . . . . 6 (𝐶 ∈ ({1𝑜} × 𝐵) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
26 xp1st 7347 . . . . . . 7 (𝐶 ∈ ({1𝑜} × 𝐵) → (1st𝐶) ∈ {1𝑜})
27 elsni 4333 . . . . . . 7 ((1st𝐶) ∈ {1𝑜} → (1st𝐶) = 1𝑜)
28 opeq1 4539 . . . . . . . 8 ((1st𝐶) = 1𝑜 → ⟨(1st𝐶), (2nd𝐶)⟩ = ⟨1𝑜, (2nd𝐶)⟩)
2928eqeq2d 2781 . . . . . . 7 ((1st𝐶) = 1𝑜 → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨1𝑜, (2nd𝐶)⟩))
3026, 27, 293syl 18 . . . . . 6 (𝐶 ∈ ({1𝑜} × 𝐵) → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨1𝑜, (2nd𝐶)⟩))
3125, 30mpbid 222 . . . . 5 (𝐶 ∈ ({1𝑜} × 𝐵) → 𝐶 = ⟨1𝑜, (2nd𝐶)⟩)
32 fvexd 6344 . . . . . 6 (𝐶 ∈ ({1𝑜} × 𝐵) → (2nd𝐶) ∈ V)
33 opex 5060 . . . . . 6 ⟨1𝑜, (2nd𝐶)⟩ ∈ V
34 opeq2 4540 . . . . . . 7 (𝑧 = (2nd𝐶) → ⟨1𝑜, 𝑧⟩ = ⟨1𝑜, (2nd𝐶)⟩)
35 df-inr 8930 . . . . . . 7 inr = (𝑧 ∈ V ↦ ⟨1𝑜, 𝑧⟩)
3634, 35fvmptg 6422 . . . . . 6 (((2nd𝐶) ∈ V ∧ ⟨1𝑜, (2nd𝐶)⟩ ∈ V) → (inr‘(2nd𝐶)) = ⟨1𝑜, (2nd𝐶)⟩)
3732, 33, 36sylancl 574 . . . . 5 (𝐶 ∈ ({1𝑜} × 𝐵) → (inr‘(2nd𝐶)) = ⟨1𝑜, (2nd𝐶)⟩)
3831, 37eqtr4d 2808 . . . 4 (𝐶 ∈ ({1𝑜} × 𝐵) → 𝐶 = (inr‘(2nd𝐶)))
39 fveq2 6332 . . . . . 6 (𝑥 = (2nd𝐶) → (inr‘𝑥) = (inr‘(2nd𝐶)))
4039eqeq2d 2781 . . . . 5 (𝑥 = (2nd𝐶) → (𝐶 = (inr‘𝑥) ↔ 𝐶 = (inr‘(2nd𝐶))))
4140rspcev 3460 . . . 4 (((2nd𝐶) ∈ 𝐵𝐶 = (inr‘(2nd𝐶))) → ∃𝑥𝐵 𝐶 = (inr‘𝑥))
4224, 38, 41syl2anc 573 . . 3 (𝐶 ∈ ({1𝑜} × 𝐵) → ∃𝑥𝐵 𝐶 = (inr‘𝑥))
4323, 42orim12i 892 . 2 ((𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1𝑜} × 𝐵)) → (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
444, 43syl 17 1 (𝐶 ∈ (𝐴𝐵) → (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 834   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  Vcvv 3351   ∪ cun 3721  ∅c0 4063  {csn 4316  ⟨cop 4322   × cxp 5247  ‘cfv 6031  1st c1st 7313  2nd c2nd 7314  1𝑜c1o 7706   ⊔ cdju 8925  inlcinl 8926  inrcinr 8927 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7315  df-2nd 7316  df-dju 8928  df-inl 8929  df-inr 8930 This theorem is referenced by:  djuss  8946  djuun  8952  updjud  8960
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