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Mirrors > Home > ILE Home > Th. List > djulclr | GIF version |
Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.) |
Ref | Expression |
---|---|
djulclr | ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 5504 | . 2 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) = (inl‘𝐶)) | |
2 | elex 2732 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
3 | 0ex 4103 | . . . . . 6 ⊢ ∅ ∈ V | |
4 | 3 | snid 3601 | . . . . 5 ⊢ ∅ ∈ {∅} |
5 | opelxpi 4630 | . . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴) → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) | |
6 | 4, 5 | mpan 421 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) |
7 | opeq2 3753 | . . . . 5 ⊢ (𝑥 = 𝐶 → 〈∅, 𝑥〉 = 〈∅, 𝐶〉) | |
8 | df-inl 7003 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
9 | 7, 8 | fvmptg 5556 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) → (inl‘𝐶) = 〈∅, 𝐶〉) |
10 | 2, 6, 9 | syl2anc 409 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) = 〈∅, 𝐶〉) |
11 | elun1 3284 | . . . . 5 ⊢ (〈∅, 𝐶〉 ∈ ({∅} × 𝐴) → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
12 | 6, 11 | syl 14 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
13 | df-dju 6994 | . . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
14 | 12, 13 | eleqtrrdi 2258 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
15 | 10, 14 | eqeltrd 2241 | . 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
16 | 1, 15 | eqeltrd 2241 | 1 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∪ cun 3109 ∅c0 3404 {csn 3570 〈cop 3573 × cxp 4596 ↾ cres 4600 ‘cfv 5182 1oc1o 6368 ⊔ cdju 6993 inlcinl 7001 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-res 4610 df-iota 5147 df-fun 5184 df-fv 5190 df-dju 6994 df-inl 7003 |
This theorem is referenced by: inlresf1 7017 |
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