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Theorem djur 9340
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 9322 . . . 4 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2902 . . 3 (𝐶 ∈ (𝐴𝐵) ↔ 𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 4123 . . 3 (𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
42, 3sylbb 221 . 2 (𝐶 ∈ (𝐴𝐵) → (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
5 xp2nd 7714 . . . 4 (𝐶 ∈ ({∅} × 𝐴) → (2nd𝐶) ∈ 𝐴)
6 1st2nd2 7720 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
7 xp1st 7713 . . . . . . 7 (𝐶 ∈ ({∅} × 𝐴) → (1st𝐶) ∈ {∅})
8 elsni 4576 . . . . . . 7 ((1st𝐶) ∈ {∅} → (1st𝐶) = ∅)
9 opeq1 4795 . . . . . . . 8 ((1st𝐶) = ∅ → ⟨(1st𝐶), (2nd𝐶)⟩ = ⟨∅, (2nd𝐶)⟩)
109eqeq2d 2830 . . . . . . 7 ((1st𝐶) = ∅ → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd𝐶)⟩))
117, 8, 103syl 18 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd𝐶)⟩))
126, 11mpbid 234 . . . . 5 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨∅, (2nd𝐶)⟩)
13 fvexd 6678 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → (2nd𝐶) ∈ V)
14 opex 5347 . . . . . 6 ⟨∅, (2nd𝐶)⟩ ∈ V
15 opeq2 4796 . . . . . . 7 (𝑦 = (2nd𝐶) → ⟨∅, 𝑦⟩ = ⟨∅, (2nd𝐶)⟩)
16 df-inl 9323 . . . . . . 7 inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
1715, 16fvmptg 6759 . . . . . 6 (((2nd𝐶) ∈ V ∧ ⟨∅, (2nd𝐶)⟩ ∈ V) → (inl‘(2nd𝐶)) = ⟨∅, (2nd𝐶)⟩)
1813, 14, 17sylancl 588 . . . . 5 (𝐶 ∈ ({∅} × 𝐴) → (inl‘(2nd𝐶)) = ⟨∅, (2nd𝐶)⟩)
1912, 18eqtr4d 2857 . . . 4 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = (inl‘(2nd𝐶)))
20 fveq2 6663 . . . . 5 (𝑥 = (2nd𝐶) → (inl‘𝑥) = (inl‘(2nd𝐶)))
2120rspceeqv 3636 . . . 4 (((2nd𝐶) ∈ 𝐴𝐶 = (inl‘(2nd𝐶))) → ∃𝑥𝐴 𝐶 = (inl‘𝑥))
225, 19, 21syl2anc 586 . . 3 (𝐶 ∈ ({∅} × 𝐴) → ∃𝑥𝐴 𝐶 = (inl‘𝑥))
23 xp2nd 7714 . . . 4 (𝐶 ∈ ({1o} × 𝐵) → (2nd𝐶) ∈ 𝐵)
24 1st2nd2 7720 . . . . . 6 (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
25 xp1st 7713 . . . . . . 7 (𝐶 ∈ ({1o} × 𝐵) → (1st𝐶) ∈ {1o})
26 elsni 4576 . . . . . . 7 ((1st𝐶) ∈ {1o} → (1st𝐶) = 1o)
27 opeq1 4795 . . . . . . . 8 ((1st𝐶) = 1o → ⟨(1st𝐶), (2nd𝐶)⟩ = ⟨1o, (2nd𝐶)⟩)
2827eqeq2d 2830 . . . . . . 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 234 . . . . 5 (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨1o, (2nd𝐶)⟩)
31 fvexd 6678 . . . . . 6 (𝐶 ∈ ({1o} × 𝐵) → (2nd𝐶) ∈ V)
32 opex 5347 . . . . . 6 ⟨1o, (2nd𝐶)⟩ ∈ V
33 opeq2 4796 . . . . . . 7 (𝑧 = (2nd𝐶) → ⟨1o, 𝑧⟩ = ⟨1o, (2nd𝐶)⟩)
34 df-inr 9324 . . . . . . 7 inr = (𝑧 ∈ V ↦ ⟨1o, 𝑧⟩)
3533, 34fvmptg 6759 . . . . . 6 (((2nd𝐶) ∈ V ∧ ⟨1o, (2nd𝐶)⟩ ∈ V) → (inr‘(2nd𝐶)) = ⟨1o, (2nd𝐶)⟩)
3631, 32, 35sylancl 588 . . . . 5 (𝐶 ∈ ({1o} × 𝐵) → (inr‘(2nd𝐶)) = ⟨1o, (2nd𝐶)⟩)
3730, 36eqtr4d 2857 . . . 4 (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = (inr‘(2nd𝐶)))
38 fveq2 6663 . . . . 5 (𝑥 = (2nd𝐶) → (inr‘𝑥) = (inr‘(2nd𝐶)))
3938rspceeqv 3636 . . . 4 (((2nd𝐶) ∈ 𝐵𝐶 = (inr‘(2nd𝐶))) → ∃𝑥𝐵 𝐶 = (inr‘𝑥))
4023, 37, 39syl2anc 586 . . 3 (𝐶 ∈ ({1o} × 𝐵) → ∃𝑥𝐵 𝐶 = (inr‘𝑥))
4122, 40orim12i 905 . 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 208  wo 843   = wceq 1531  wcel 2108  wrex 3137  Vcvv 3493  cun 3932  c0 4289  {csn 4559  cop 4565   × cxp 5546  cfv 6348  1st c1st 7679  2nd c2nd 7680  1oc1o 8087  cdju 9319  inlcinl 9320  inrcinr 9321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-1st 7681  df-2nd 7682  df-dju 9322  df-inl 9323  df-inr 9324
This theorem is referenced by:  djuss  9341  djuun  9347  updjud  9355
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