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Mirrors > Home > ILE Home > Th. List > djurcl | GIF version |
Description: Right closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djurcl | ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2750 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ V) | |
2 | 1oex 6427 | . . . . 5 ⊢ 1o ∈ V | |
3 | 2 | snid 3625 | . . . 4 ⊢ 1o ∈ {1o} |
4 | opelxpi 4660 | . . . 4 ⊢ ((1o ∈ {1o} ∧ 𝐶 ∈ 𝐵) → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) | |
5 | 3, 4 | mpan 424 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) |
6 | opeq2 3781 | . . . 4 ⊢ (𝑥 = 𝐶 → 〈1o, 𝑥〉 = 〈1o, 𝐶〉) | |
7 | df-inr 7049 | . . . 4 ⊢ inr = (𝑥 ∈ V ↦ 〈1o, 𝑥〉) | |
8 | 6, 7 | fvmptg 5594 | . . 3 ⊢ ((𝐶 ∈ V ∧ 〈1o, 𝐶〉 ∈ ({1o} × 𝐵)) → (inr‘𝐶) = 〈1o, 𝐶〉) |
9 | 1, 5, 8 | syl2anc 411 | . 2 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) = 〈1o, 𝐶〉) |
10 | elun2 3305 | . . . 4 ⊢ (〈1o, 𝐶〉 ∈ ({1o} × 𝐵) → 〈1o, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
11 | 5, 10 | syl 14 | . . 3 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
12 | df-dju 7039 | . . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
13 | 11, 12 | eleqtrrdi 2271 | . 2 ⊢ (𝐶 ∈ 𝐵 → 〈1o, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
14 | 9, 13 | eqeltrd 2254 | 1 ⊢ (𝐶 ∈ 𝐵 → (inr‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2739 ∪ cun 3129 ∅c0 3424 {csn 3594 〈cop 3597 × cxp 4626 ‘cfv 5218 1oc1o 6412 ⊔ cdju 7038 inrcinr 7047 |
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 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 |
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 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-1o 6419 df-dju 7039 df-inr 7049 |
This theorem is referenced by: updjudhcoinrg 7082 omp1eomlem 7095 difinfsnlem 7100 difinfsn 7101 0ct 7108 ctmlemr 7109 ctssdclemn0 7111 exmidfodomrlemr 7203 exmidfodomrlemrALT 7204 |
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