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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-xpcossxp | Structured version Visualization version GIF version |
Description: The composition of two Cartesian products is included in the expected Cartesian product. There is equality if (𝐵 ∩ 𝐶) ≠ ∅, see xpcogend 14939. (Contributed by BJ, 22-May-2024.) |
Ref | Expression |
---|---|
bj-xpcossxp | ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brxp 5721 | . . . . . . 7 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | brxp 5721 | . . . . . . 7 ⊢ (𝑦(𝐶 × 𝐷)𝑡 ↔ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷)) | |
3 | 1, 2 | anbi12i 626 | . . . . . 6 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷))) |
4 | an43 657 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
5 | 3, 4 | bitri 275 | . . . . 5 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
6 | 5 | exbii 1843 | . . . 4 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
7 | 19.42v 1950 | . . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
8 | 7 | simplbi 497 | . . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)) |
9 | 6, 8 | sylbi 216 | . . 3 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) → (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)) |
10 | 9 | ssopab2i 5546 | . 2 ⊢ {⟨𝑥, 𝑡⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡)} ⊆ {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)} |
11 | df-co 5681 | . 2 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑡⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡)} | |
12 | df-xp 5678 | . 2 ⊢ (𝐴 × 𝐷) = {⟨𝑥, 𝑡⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)} | |
13 | 10, 11, 12 | 3sstr4i 4021 | 1 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∃wex 1774 ∈ wcel 2099 ⊆ wss 3944 class class class wbr 5142 {copab 5204 × cxp 5670 ∘ ccom 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-xp 5678 df-co 5681 |
This theorem is referenced by: bj-imdirco 36592 |
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