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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 5558 | . 2 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) = (inl‘𝐶)) | |
2 | elex 2763 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
3 | 0ex 4145 | . . . . . 6 ⊢ ∅ ∈ V | |
4 | 3 | snid 3638 | . . . . 5 ⊢ ∅ ∈ {∅} |
5 | opelxpi 4676 | . . . . 5 ⊢ ((∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴) → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) | |
6 | 4, 5 | mpan 424 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) |
7 | opeq2 3794 | . . . . 5 ⊢ (𝑥 = 𝐶 → 〈∅, 𝑥〉 = 〈∅, 𝐶〉) | |
8 | df-inl 7076 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
9 | 7, 8 | fvmptg 5613 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) → (inl‘𝐶) = 〈∅, 𝐶〉) |
10 | 2, 6, 9 | syl2anc 411 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) = 〈∅, 𝐶〉) |
11 | elun1 3317 | . . . . 5 ⊢ (〈∅, 𝐶〉 ∈ ({∅} × 𝐴) → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
12 | 6, 11 | syl 14 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
13 | df-dju 7067 | . . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
14 | 12, 13 | eleqtrrdi 2283 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
15 | 10, 14 | eqeltrd 2266 | . 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
16 | 1, 15 | eqeltrd 2266 | 1 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 ∪ cun 3142 ∅c0 3437 {csn 3607 〈cop 3610 × cxp 4642 ↾ cres 4646 ‘cfv 5235 1oc1o 6434 ⊔ cdju 7066 inlcinl 7074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-res 4656 df-iota 5196 df-fun 5237 df-fv 5243 df-dju 7067 df-inl 7076 |
This theorem is referenced by: inlresf1 7090 |
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