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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 5397 | . 2 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) = (inl‘𝐶)) | |
2 | elex 2666 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
3 | 0ex 4013 | . . . . . 6 ⊢ ∅ ∈ V | |
4 | 3 | snid 3520 | . . . . 5 ⊢ ∅ ∈ {∅} |
5 | opelxpi 4529 | . . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴) → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) | |
6 | 4, 5 | mpan 418 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) |
7 | opeq2 3670 | . . . . 5 ⊢ (𝑥 = 𝐶 → 〈∅, 𝑥〉 = 〈∅, 𝐶〉) | |
8 | df-inl 6882 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
9 | 7, 8 | fvmptg 5449 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) → (inl‘𝐶) = 〈∅, 𝐶〉) |
10 | 2, 6, 9 | syl2anc 406 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) = 〈∅, 𝐶〉) |
11 | elun1 3207 | . . . . 5 ⊢ (〈∅, 𝐶〉 ∈ ({∅} × 𝐴) → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
12 | 6, 11 | syl 14 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
13 | df-dju 6873 | . . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
14 | 12, 13 | syl6eleqr 2206 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
15 | 10, 14 | eqeltrd 2189 | . 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
16 | 1, 15 | eqeltrd 2189 | 1 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 ∈ wcel 1461 Vcvv 2655 ∪ cun 3033 ∅c0 3327 {csn 3491 〈cop 3494 × cxp 4495 ↾ cres 4499 ‘cfv 5079 1oc1o 6258 ⊔ cdju 6872 inlcinl 6880 |
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 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-v 2657 df-sbc 2877 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-res 4509 df-iota 5044 df-fun 5081 df-fv 5087 df-dju 6873 df-inl 6882 |
This theorem is referenced by: inlresf1 6896 |
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