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Mirrors > Home > ILE Home > Th. List > txbasex | GIF version |
Description: The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
txval.1 | ⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) |
Ref | Expression |
---|---|
txbasex | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | txval.1 | . . . 4 ⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) | |
2 | eqid 2139 | . . . 4 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
3 | eqid 2139 | . . . 4 ⊢ ∪ 𝑆 = ∪ 𝑆 | |
4 | 1, 2, 3 | txuni2 12430 | . . 3 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ 𝐵 |
5 | uniexg 4361 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑅 ∈ V) | |
6 | uniexg 4361 | . . . 4 ⊢ (𝑆 ∈ 𝑊 → ∪ 𝑆 ∈ V) | |
7 | xpexg 4653 | . . . 4 ⊢ ((∪ 𝑅 ∈ V ∧ ∪ 𝑆 ∈ V) → (∪ 𝑅 × ∪ 𝑆) ∈ V) | |
8 | 5, 6, 7 | syl2an 287 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∪ 𝑅 × ∪ 𝑆) ∈ V) |
9 | 4, 8 | eqeltrrid 2227 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ∪ 𝐵 ∈ V) |
10 | uniexb 4394 | . 2 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
11 | 9, 10 | sylibr 133 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐵 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ∪ cuni 3736 × cxp 4537 ran crn 4540 ∈ cmpo 5776 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: txbas 12432 eltx 12433 txtopon 12436 txopn 12439 txss12 12440 txbasval 12441 txrest 12450 |
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