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Mirrors > Home > ILE Home > Th. List > djulcl | GIF version |
Description: Left closure of disjoint union. (Contributed by Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djulcl | ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2748 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V) | |
2 | 0ex 4127 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | snid 3622 | . . . 4 ⊢ ∅ ∈ {∅} |
4 | opelxpi 4655 | . . . 4 ⊢ ((∅ ∈ {∅} ∧ 𝐶 ∈ 𝐴) → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) | |
5 | 3, 4 | mpan 424 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) |
6 | opeq2 3777 | . . . 4 ⊢ (𝑥 = 𝐶 → 〈∅, 𝑥〉 = 〈∅, 𝐶〉) | |
7 | df-inl 7040 | . . . 4 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
8 | 6, 7 | fvmptg 5588 | . . 3 ⊢ ((𝐶 ∈ V ∧ 〈∅, 𝐶〉 ∈ ({∅} × 𝐴)) → (inl‘𝐶) = 〈∅, 𝐶〉) |
9 | 1, 5, 8 | syl2anc 411 | . 2 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) = 〈∅, 𝐶〉) |
10 | elun1 3302 | . . . 4 ⊢ (〈∅, 𝐶〉 ∈ ({∅} × 𝐴) → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) | |
11 | 5, 10 | syl 14 | . . 3 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
12 | df-dju 7031 | . . 3 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
13 | 11, 12 | eleqtrrdi 2271 | . 2 ⊢ (𝐶 ∈ 𝐴 → 〈∅, 𝐶〉 ∈ (𝐴 ⊔ 𝐵)) |
14 | 9, 13 | eqeltrd 2254 | 1 ⊢ (𝐶 ∈ 𝐴 → (inl‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∪ cun 3127 ∅c0 3422 {csn 3591 〈cop 3594 × cxp 4621 ‘cfv 5212 1oc1o 6404 ⊔ cdju 7030 inlcinl 7038 |
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-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-dju 7031 df-inl 7040 |
This theorem is referenced by: djulclb 7048 updjudhcoinlf 7073 omp1eomlem 7087 difinfsnlem 7092 difinfsn 7093 ctmlemr 7101 ctm 7102 ctssdclemn0 7103 ctssdccl 7104 fodju0 7139 exmidfodomrlemr 7195 exmidfodomrlemrALT 7196 subctctexmid 14399 |
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