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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 14325. (Contributed by BJ, 22-May-2024.) |
Ref | Expression |
---|---|
bj-xpcossxp | ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brxp 5565 | . . . . . . 7 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | brxp 5565 | . . . . . . 7 ⊢ (𝑦(𝐶 × 𝐷)𝑡 ↔ (𝑦 ∈ 𝐶 ∧ 𝑡 ∈ 𝐷)) | |
3 | 1, 2 | anbi12i 629 | . . . . . 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 278 | . . . . 5 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
6 | 5 | exbii 1849 | . . . 4 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
7 | 19.42v 1954 | . . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
8 | 7 | simplbi 501 | . . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) → (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)) |
9 | 6, 8 | sylbi 220 | . . 3 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡) → (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)) |
10 | 9 | ssopab2i 5402 | . 2 ⊢ {〈𝑥, 𝑡〉 ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡)} ⊆ {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)} |
11 | df-co 5528 | . 2 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {〈𝑥, 𝑡〉 ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑡)} | |
12 | df-xp 5525 | . 2 ⊢ (𝐴 × 𝐷) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 ∈ 𝐷)} | |
13 | 10, 11, 12 | 3sstr4i 3958 | 1 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∃wex 1781 ∈ wcel 2111 ⊆ wss 3881 class class class wbr 5030 {copab 5092 × cxp 5517 ∘ ccom 5523 |
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 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
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-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-co 5528 |
This theorem is referenced by: bj-imdirco 34605 |
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