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Mirrors > Home > ILE Home > Th. List > xpexgALT | GIF version |
Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4661 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
xpexgALT | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunid 3876 | . . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 {𝑦} = 𝐵 | |
2 | 1 | xpeq2i 4568 | . . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = (𝐴 × 𝐵) |
3 | xpiundi 4605 | . . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) | |
4 | 2, 3 | eqtr3i 2163 | . 2 ⊢ (𝐴 × 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) |
5 | id 19 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑊) | |
6 | fconstmpt 4594 | . . . . 5 ⊢ (𝐴 × {𝑦}) = (𝑥 ∈ 𝐴 ↦ 𝑦) | |
7 | mptexg 5653 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝑦) ∈ V) | |
8 | 6, 7 | eqeltrid 2227 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {𝑦}) ∈ V) |
9 | 8 | ralrimivw 2509 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) |
10 | iunexg 6025 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) | |
11 | 5, 9, 10 | syl2anr 288 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) |
12 | 4, 11 | eqeltrid 2227 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 ∀wral 2417 Vcvv 2689 {csn 3532 ∪ ciun 3821 ↦ cmpt 3997 × cxp 4545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 |
This theorem is referenced by: (None) |
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