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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 5049 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
2 | snssi 4812 | . . 3 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
3 | ssv 4007 | . . 3 ⊢ 𝐵 ⊆ V | |
4 | xpss12 5692 | . . 3 ⊢ (({𝑥} ⊆ 𝐴 ∧ 𝐵 ⊆ V) → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) | |
5 | 2, 3, 4 | sylancl 587 | . 2 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × 𝐵) ⊆ (𝐴 × V)) |
6 | 1, 5 | mprgbir 3069 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 ⊆ wss 3949 {csn 4629 ∪ ciun 4998 × cxp 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-v 3477 df-in 3956 df-ss 3966 df-sn 4630 df-iun 5000 df-opab 5212 df-xp 5683 |
This theorem is referenced by: djudisj 6167 iundom2g 10535 |
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