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Theorem djuss 7047
Description: A disjoint union is a subset of a Cartesian product. (Contributed by AV, 25-Jun-2022.)
Assertion
Ref Expression
djuss (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))

Proof of Theorem djuss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 djur 7046 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)))
2 simpr 109 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = (inl‘𝑦))
3 df-inl 7024 . . . . . . . . 9 inl = (𝑥 ∈ V ↦ ⟨∅, 𝑥⟩)
4 opeq2 3766 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨∅, 𝑥⟩ = ⟨∅, 𝑦⟩)
5 elex 2741 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ V)
6 0ex 4116 . . . . . . . . . . 11 ∅ ∈ V
7 vex 2733 . . . . . . . . . . 11 𝑦 ∈ V
86, 7opex 4214 . . . . . . . . . 10 ⟨∅, 𝑦⟩ ∈ V
98a1i 9 . . . . . . . . 9 (𝑦𝐴 → ⟨∅, 𝑦⟩ ∈ V)
103, 4, 5, 9fvmptd3 5589 . . . . . . . 8 (𝑦𝐴 → (inl‘𝑦) = ⟨∅, 𝑦⟩)
1110adantr 274 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (inl‘𝑦) = ⟨∅, 𝑦⟩)
122, 11eqtrd 2203 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 = ⟨∅, 𝑦⟩)
13 elun1 3294 . . . . . . . . 9 (𝑦𝐴𝑦 ∈ (𝐴𝐵))
146prid1 3689 . . . . . . . . 9 ∅ ∈ {∅, 1o}
1513, 14jctil 310 . . . . . . . 8 (𝑦𝐴 → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1615adantr 274 . . . . . . 7 ((𝑦𝐴𝑥 = (inl‘𝑦)) → (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
17 opelxp 4641 . . . . . . 7 (⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (∅ ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
1816, 17sylibr 133 . . . . . 6 ((𝑦𝐴𝑥 = (inl‘𝑦)) → ⟨∅, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
1912, 18eqeltrd 2247 . . . . 5 ((𝑦𝐴𝑥 = (inl‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
2019rexlimiva 2582 . . . 4 (∃𝑦𝐴 𝑥 = (inl‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
21 simpr 109 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = (inr‘𝑦))
22 df-inr 7025 . . . . . . . . 9 inr = (𝑥 ∈ V ↦ ⟨1o, 𝑥⟩)
23 opeq2 3766 . . . . . . . . 9 (𝑥 = 𝑦 → ⟨1o, 𝑥⟩ = ⟨1o, 𝑦⟩)
24 elex 2741 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ V)
25 1oex 6403 . . . . . . . . . . 11 1o ∈ V
2625, 7opex 4214 . . . . . . . . . 10 ⟨1o, 𝑦⟩ ∈ V
2726a1i 9 . . . . . . . . 9 (𝑦𝐵 → ⟨1o, 𝑦⟩ ∈ V)
2822, 23, 24, 27fvmptd3 5589 . . . . . . . 8 (𝑦𝐵 → (inr‘𝑦) = ⟨1o, 𝑦⟩)
2928adantr 274 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (inr‘𝑦) = ⟨1o, 𝑦⟩)
3021, 29eqtrd 2203 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 = ⟨1o, 𝑦⟩)
31 elun2 3295 . . . . . . . . 9 (𝑦𝐵𝑦 ∈ (𝐴𝐵))
3231adantr 274 . . . . . . . 8 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑦 ∈ (𝐴𝐵))
3325prid2 3690 . . . . . . . 8 1o ∈ {∅, 1o}
3432, 33jctil 310 . . . . . . 7 ((𝑦𝐵𝑥 = (inr‘𝑦)) → (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
35 opelxp 4641 . . . . . . 7 (⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)) ↔ (1o ∈ {∅, 1o} ∧ 𝑦 ∈ (𝐴𝐵)))
3634, 35sylibr 133 . . . . . 6 ((𝑦𝐵𝑥 = (inr‘𝑦)) → ⟨1o, 𝑦⟩ ∈ ({∅, 1o} × (𝐴𝐵)))
3730, 36eqeltrd 2247 . . . . 5 ((𝑦𝐵𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3837rexlimiva 2582 . . . 4 (∃𝑦𝐵 𝑥 = (inr‘𝑦) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
3920, 38jaoi 711 . . 3 ((∃𝑦𝐴 𝑥 = (inl‘𝑦) ∨ ∃𝑦𝐵 𝑥 = (inr‘𝑦)) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
401, 39sylbi 120 . 2 (𝑥 ∈ (𝐴𝐵) → 𝑥 ∈ ({∅, 1o} × (𝐴𝐵)))
4140ssriv 3151 1 (𝐴𝐵) ⊆ ({∅, 1o} × (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wo 703   = wceq 1348  wcel 2141  wrex 2449  Vcvv 2730  cun 3119  wss 3121  c0 3414  {cpr 3584  cop 3586   × cxp 4609  cfv 5198  1oc1o 6388  cdju 7014  inlcinl 7022  inrcinr 7023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dju 7015  df-inl 7024  df-inr 7025
This theorem is referenced by:  eldju1st  7048
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