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Mirrors > Home > ILE Home > Th. List > Mathboxes > djulclALT | GIF version |
Description: Shortening of djulcl 7043 using djucllem 14174. (Contributed by BJ, 4-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
djulclALT | ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4127 | . . . . 5 ⊢ ∅ ∈ V | |
2 | df-inl 7039 | . . . . 5 ⊢ inl = (𝑥 ∈ V ↦ 〈∅, 𝑥〉) | |
3 | 1, 2 | djucllem 14174 | . . . 4 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ ({∅} × 𝐴)) |
4 | 3 | orcd 733 | . . 3 ⊢ (𝐶 ∈ 𝐴 → (((inl ↾ 𝐴)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inl ↾ 𝐴)‘𝐶) ∈ ({1o} × 𝐵))) |
5 | elun 3276 | . . 3 ⊢ (((inl ↾ 𝐴)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵)) ↔ (((inl ↾ 𝐴)‘𝐶) ∈ ({∅} × 𝐴) ∨ ((inl ↾ 𝐴)‘𝐶) ∈ ({1o} × 𝐵))) | |
6 | 4, 5 | sylibr 134 | . 2 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (({∅} × 𝐴) ∪ ({1o} × 𝐵))) |
7 | df-dju 7030 | . 2 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1o} × 𝐵)) | |
8 | 6, 7 | eleqtrrdi 2271 | 1 ⊢ (𝐶 ∈ 𝐴 → ((inl ↾ 𝐴)‘𝐶) ∈ (𝐴 ⊔ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 708 ∈ wcel 2148 ∪ cun 3127 ∅c0 3422 {csn 3591 × cxp 4620 ↾ cres 4624 ‘cfv 5211 1oc1o 6403 ⊔ cdju 7029 inlcinl 7037 |
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 4205 |
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 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-res 4634 df-iota 5173 df-fun 5213 df-fv 5219 df-dju 7030 df-inl 7039 |
This theorem is referenced by: (None) |
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