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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 5445 | . 2 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) = (inr‘𝐶)) | |
2 | elex 2697 | . . . 4 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V) | |
3 | 1oex 6321 | . . . . . 6 ⊢ 1o ∈ V | |
4 | 3 | snid 3556 | . . . . 5 ⊢ 1o ∈ {1o} |
5 | opelxpi 4571 | . . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) | |
6 | 4, 5 | mpan 420 | . . . 4 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) |
7 | opeq2 3706 | . . . . 5 ⊢ (𝑥 = 𝐶 → 〈1o, 𝑥〉 = 〈1o, 𝐶〉) | |
8 | df-inr 6933 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
9 | 7, 8 | fvmptg 5497 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) → (inr‘𝐶) = 〈1o, 𝐶〉) |
10 | 2, 6, 9 | syl2anc 408 | . . 3 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) = 〈1o, 𝐶〉) |
11 | elun2 3244 | . . . . 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 6923 | . . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
14 | 12, 13 | eleqtrrdi 2233 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
15 | 10, 14 | eqeltrd 2216 | . 2 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
16 | 1, 15 | eqeltrd 2216 | 1 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cun 3069 ∅c0 3363 {csn 3527 〈cop 3530 × cxp 4537 ↾ cres 4541 ‘cfv 5123 1oc1o 6306 ⊔ cdju 6922 inrcinr 6931 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 df-1o 6313 df-dju 6923 df-inr 6933 |
This theorem is referenced by: inrresf1 6947 |
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