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Mirrors > Home > MPE Home > Th. List > txbasex | Structured version Visualization version 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 2798 | . . . 4 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
3 | eqid 2798 | . . . 4 ⊢ ∪ 𝑆 = ∪ 𝑆 | |
4 | 1, 2, 3 | txuni2 22170 | . . 3 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ 𝐵 |
5 | uniexg 7446 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑅 ∈ V) | |
6 | uniexg 7446 | . . . 4 ⊢ (𝑆 ∈ 𝑊 → ∪ 𝑆 ∈ V) | |
7 | xpexg 7453 | . . . 4 ⊢ ((∪ 𝑅 ∈ V ∧ ∪ 𝑆 ∈ V) → (∪ 𝑅 × ∪ 𝑆) ∈ V) | |
8 | 5, 6, 7 | syl2an 598 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∪ 𝑅 × ∪ 𝑆) ∈ V) |
9 | 4, 8 | eqeltrrid 2895 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ∪ 𝐵 ∈ V) |
10 | uniexb 7466 | . 2 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
11 | 9, 10 | sylibr 237 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cuni 4800 × cxp 5517 ran crn 5520 ∈ cmpo 7137 |
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-pow 5231 ax-pr 5295 ax-un 7441 |
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-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: txbas 22172 eltx 22173 txtopon 22196 txopn 22207 txss12 22210 txbasval 22211 txrest 22236 sxsiga 31560 elsx 31563 mbfmco2 31633 |
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