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Mirrors > Home > ILE Home > Th. List > djurclr | GIF version |
Description: Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) (Revised by BJ, 6-Jul-2022.) |
Ref | Expression |
---|---|
djurclr | ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 5453 | . 2 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) = (inr‘𝐶)) | |
2 | elex 2700 | . . . 4 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V) | |
3 | 1oex 6329 | . . . . . 6 ⊢ 1o ∈ V | |
4 | 3 | snid 3563 | . . . . 5 ⊢ 1o ∈ {1o} |
5 | opelxpi 4579 | . . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) | |
6 | 4, 5 | mpan 421 | . . . 4 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) |
7 | opeq2 3714 | . . . . 5 ⊢ (𝑥 = 𝐶 → 〈1o, 𝑥〉 = 〈1o, 𝐶〉) | |
8 | df-inr 6941 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
9 | 7, 8 | fvmptg 5505 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) → (inr‘𝐶) = 〈1o, 𝐶〉) |
10 | 2, 6, 9 | syl2anc 409 | . . 3 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) = 〈1o, 𝐶〉) |
11 | elun2 3249 | . . . . 5 ⊢ (〈1o, 𝐶〉 ∈ ({1o} × 𝐵) → 〈1o, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
12 | 6, 11 | syl 14 | . . . 4 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
13 | df-dju 6931 | . . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
14 | 12, 13 | eleqtrrdi 2234 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
15 | 10, 14 | eqeltrd 2217 | . 2 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
16 | 1, 15 | eqeltrd 2217 | 1 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∪ cun 3074 ∅c0 3368 {csn 3532 〈cop 3535 × cxp 4545 ↾ cres 4549 ‘cfv 5131 1oc1o 6314 ⊔ cdju 6930 inrcinr 6939 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-res 4559 df-iota 5096 df-fun 5133 df-fv 5139 df-1o 6321 df-dju 6931 df-inr 6941 |
This theorem is referenced by: inrresf1 6955 |
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