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Theorem djur 6837
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 6811 . . . 4 (𝐴𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵))
21eleq2i 2161 . . 3 (𝐶 ∈ (𝐴𝐵) ↔ 𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)))
3 elun 3156 . . 3 (𝐶 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
42, 3sylbb 122 . 2 (𝐶 ∈ (𝐴𝐵) → (𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)))
5 xp2nd 5975 . . . 4 (𝐶 ∈ ({∅} × 𝐴) → (2nd𝐶) ∈ 𝐴)
6 1st2nd2 5983 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
7 xp1st 5974 . . . . . . 7 (𝐶 ∈ ({∅} × 𝐴) → (1st𝐶) ∈ {∅})
8 elsni 3484 . . . . . . 7 ((1st𝐶) ∈ {∅} → (1st𝐶) = ∅)
9 opeq1 3644 . . . . . . . 8 ((1st𝐶) = ∅ → ⟨(1st𝐶), (2nd𝐶)⟩ = ⟨∅, (2nd𝐶)⟩)
109eqeq2d 2106 . . . . . . 7 ((1st𝐶) = ∅ → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd𝐶)⟩))
117, 8, 103syl 17 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨∅, (2nd𝐶)⟩))
126, 11mpbid 146 . . . . 5 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = ⟨∅, (2nd𝐶)⟩)
13 2ndexg 5977 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → (2nd𝐶) ∈ V)
14 0ex 3987 . . . . . . 7 ∅ ∈ V
15 opexg 4079 . . . . . . 7 ((∅ ∈ V ∧ (2nd𝐶) ∈ V) → ⟨∅, (2nd𝐶)⟩ ∈ V)
1614, 13, 15sylancr 406 . . . . . 6 (𝐶 ∈ ({∅} × 𝐴) → ⟨∅, (2nd𝐶)⟩ ∈ V)
17 opeq2 3645 . . . . . . 7 (𝑦 = (2nd𝐶) → ⟨∅, 𝑦⟩ = ⟨∅, (2nd𝐶)⟩)
18 df-inl 6819 . . . . . . 7 inl = (𝑦 ∈ V ↦ ⟨∅, 𝑦⟩)
1917, 18fvmptg 5415 . . . . . 6 (((2nd𝐶) ∈ V ∧ ⟨∅, (2nd𝐶)⟩ ∈ V) → (inl‘(2nd𝐶)) = ⟨∅, (2nd𝐶)⟩)
2013, 16, 19syl2anc 404 . . . . 5 (𝐶 ∈ ({∅} × 𝐴) → (inl‘(2nd𝐶)) = ⟨∅, (2nd𝐶)⟩)
2112, 20eqtr4d 2130 . . . 4 (𝐶 ∈ ({∅} × 𝐴) → 𝐶 = (inl‘(2nd𝐶)))
22 fveq2 5340 . . . . . 6 (𝑥 = (2nd𝐶) → (inl‘𝑥) = (inl‘(2nd𝐶)))
2322eqeq2d 2106 . . . . 5 (𝑥 = (2nd𝐶) → (𝐶 = (inl‘𝑥) ↔ 𝐶 = (inl‘(2nd𝐶))))
2423rspcev 2736 . . . 4 (((2nd𝐶) ∈ 𝐴𝐶 = (inl‘(2nd𝐶))) → ∃𝑥𝐴 𝐶 = (inl‘𝑥))
255, 21, 24syl2anc 404 . . 3 (𝐶 ∈ ({∅} × 𝐴) → ∃𝑥𝐴 𝐶 = (inl‘𝑥))
26 xp2nd 5975 . . . 4 (𝐶 ∈ ({1o} × 𝐵) → (2nd𝐶) ∈ 𝐵)
27 1st2nd2 5983 . . . . . 6 (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩)
28 xp1st 5974 . . . . . . 7 (𝐶 ∈ ({1o} × 𝐵) → (1st𝐶) ∈ {1o})
29 elsni 3484 . . . . . . 7 ((1st𝐶) ∈ {1o} → (1st𝐶) = 1o)
30 opeq1 3644 . . . . . . . 8 ((1st𝐶) = 1o → ⟨(1st𝐶), (2nd𝐶)⟩ = ⟨1o, (2nd𝐶)⟩)
3130eqeq2d 2106 . . . . . . 7 ((1st𝐶) = 1o → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨1o, (2nd𝐶)⟩))
3228, 29, 313syl 17 . . . . . 6 (𝐶 ∈ ({1o} × 𝐵) → (𝐶 = ⟨(1st𝐶), (2nd𝐶)⟩ ↔ 𝐶 = ⟨1o, (2nd𝐶)⟩))
3327, 32mpbid 146 . . . . 5 (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = ⟨1o, (2nd𝐶)⟩)
34 2ndexg 5977 . . . . . 6 (𝐶 ∈ ({1o} × 𝐵) → (2nd𝐶) ∈ V)
35 1onn 6319 . . . . . . 7 1o ∈ ω
36 opexg 4079 . . . . . . 7 ((1o ∈ ω ∧ (2nd𝐶) ∈ V) → ⟨1o, (2nd𝐶)⟩ ∈ V)
3735, 34, 36sylancr 406 . . . . . 6 (𝐶 ∈ ({1o} × 𝐵) → ⟨1o, (2nd𝐶)⟩ ∈ V)
38 opeq2 3645 . . . . . . 7 (𝑧 = (2nd𝐶) → ⟨1o, 𝑧⟩ = ⟨1o, (2nd𝐶)⟩)
39 df-inr 6820 . . . . . . 7 inr = (𝑧 ∈ V ↦ ⟨1o, 𝑧⟩)
4038, 39fvmptg 5415 . . . . . 6 (((2nd𝐶) ∈ V ∧ ⟨1o, (2nd𝐶)⟩ ∈ V) → (inr‘(2nd𝐶)) = ⟨1o, (2nd𝐶)⟩)
4134, 37, 40syl2anc 404 . . . . 5 (𝐶 ∈ ({1o} × 𝐵) → (inr‘(2nd𝐶)) = ⟨1o, (2nd𝐶)⟩)
4233, 41eqtr4d 2130 . . . 4 (𝐶 ∈ ({1o} × 𝐵) → 𝐶 = (inr‘(2nd𝐶)))
43 fveq2 5340 . . . . . 6 (𝑥 = (2nd𝐶) → (inr‘𝑥) = (inr‘(2nd𝐶)))
4443eqeq2d 2106 . . . . 5 (𝑥 = (2nd𝐶) → (𝐶 = (inr‘𝑥) ↔ 𝐶 = (inr‘(2nd𝐶))))
4544rspcev 2736 . . . 4 (((2nd𝐶) ∈ 𝐵𝐶 = (inr‘(2nd𝐶))) → ∃𝑥𝐵 𝐶 = (inr‘𝑥))
4626, 42, 45syl2anc 404 . . 3 (𝐶 ∈ ({1o} × 𝐵) → ∃𝑥𝐵 𝐶 = (inr‘𝑥))
4725, 46orim12i 714 . 2 ((𝐶 ∈ ({∅} × 𝐴) ∨ 𝐶 ∈ ({1o} × 𝐵)) → (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
484, 47syl 14 1 (𝐶 ∈ (𝐴𝐵) → (∃𝑥𝐴 𝐶 = (inl‘𝑥) ∨ ∃𝑥𝐵 𝐶 = (inr‘𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 667   = wceq 1296  wcel 1445  wrex 2371  Vcvv 2633  cun 3011  c0 3302  {csn 3466  cop 3469  ωcom 4433   × cxp 4465  cfv 5049  1st c1st 5947  2nd c2nd 5948  1oc1o 6212  cdju 6810  inlcinl 6817  inrcinr 6818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fo 5055  df-fv 5057  df-1st 5949  df-2nd 5950  df-1o 6219  df-dju 6811  df-inl 6819  df-inr 6820
This theorem is referenced by:  djuun  6840  djuss  6841  updjud  6853  0ct  6869  ctmlemr  6870  fodjuomnilemdc  6887  exmidfodomrlemeldju  6922
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