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Mirrors > Home > MPE Home > Th. List > djulcl | Structured version Visualization version GIF version |
Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djulcl | ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inl 9659 | . . 3 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
2 | opeq2 4811 | . . 3 ⊢ (𝑥 = 𝐶 → 〈∅, 𝑥〉 = 〈∅, 𝐶〉) | |
3 | elex 3449 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
4 | 0ex 5235 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 4 | snid 4603 | . . . 4 ⊢ ∅ ∈ {∅} |
6 | opelxpi 5626 | . . . 4 ⊢ ((∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴) → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) | |
7 | 5, 6 | mpan 687 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) |
8 | 1, 2, 3, 7 | fvmptd3 6893 | . 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) = 〈∅, 𝐶〉) |
9 | elun1 4115 | . . . 4 ⊢ (〈∅, 𝐶〉 ∈ ({∅} × 𝐴) → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
10 | 7, 9 | syl 17 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
11 | df-dju 9658 | . . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
12 | 10, 11 | eleqtrrdi 2852 | . 2 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
13 | 8, 12 | eqeltrd 2841 | 1 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Vcvv 3431 ∪ cun 3890 ∅c0 4262 {csn 4567 〈cop 4573 × cxp 5587 ‘cfv 6431 1oc1o 8279 ⊔ cdju 9655 inlcinl 9656 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6389 df-fun 6433 df-fv 6439 df-dju 9658 df-inl 9659 |
This theorem is referenced by: inlresf 9671 updjudhcoinlf 9689 |
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