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Mirrors > Home > MPE Home > Th. List > djussxp | Structured version Visualization version GIF version |
Description: Disjoint union is a subset of a Cartesian product. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Ref | Expression |
---|---|
djussxp | ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunss 4932 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
2 | snssi 4701 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
3 | ssv 3939 | . . 3 ⊢ 𝐵 ⊆ V | |
4 | xpss12 5534 | . . 3 ⊢ (({𝑥} ⊆ 𝐴 ∧ 𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
5 | 2, 3, 4 | sylancl 589 | . 2 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) |
6 | 1, 5 | mprgbir 3121 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 {csn 4525 ∪ ciun 4881 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-in 3888 df-ss 3898 df-sn 4526 df-iun 4883 df-opab 5093 df-xp 5525 |
This theorem is referenced by: djudisj 5991 iundom2g 9951 |
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