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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 5539 | . 2 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) = (inr‘𝐶)) | |
2 | elex 2748 | . . . 4 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V) | |
3 | 1oex 6424 | . . . . . 6 ⊢ 1o ∈ V | |
4 | 3 | snid 3623 | . . . . 5 ⊢ 1o ∈ {1o} |
5 | opelxpi 4658 | . . . . 5 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) | |
6 | 4, 5 | mpan 424 | . . . 4 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) |
7 | opeq2 3779 | . . . . 5 ⊢ (𝑥 = 𝐶 → 〈1o, 𝑥〉 = 〈1o, 𝐶〉) | |
8 | df-inr 7046 | . . . . 5 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
9 | 7, 8 | fvmptg 5592 | . . . 4 ⊢ ((𝐶 ∈ V ∧ 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) → (inr‘𝐶) = 〈1o, 𝐶〉) |
10 | 2, 6, 9 | syl2anc 411 | . . 3 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) = 〈1o, 𝐶〉) |
11 | elun2 3303 | . . . . 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 7036 | . . . 4 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
14 | 12, 13 | eleqtrrdi 2271 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
15 | 10, 14 | eqeltrd 2254 | . 2 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
16 | 1, 15 | eqeltrd 2254 | 1 ⊢ (𝐶 ∈ 𝐵 → ((inr ↾ 𝐵)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∪ cun 3127 ∅c0 3422 {csn 3592 〈cop 3595 × cxp 4624 ↾ cres 4628 ‘cfv 5216 1oc1o 6409 ⊔ cdju 7035 inrcinr 7044 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-res 4638 df-iota 5178 df-fun 5218 df-fv 5224 df-1o 6416 df-dju 7036 df-inr 7046 |
This theorem is referenced by: inrresf1 7060 |
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