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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 2818 | . . . 4 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
3 | eqid 2818 | . . . 4 ⊢ ∪ 𝑆 = ∪ 𝑆 | |
4 | 1, 2, 3 | txuni2 22101 | . . 3 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ 𝐵 |
5 | uniexg 7456 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑅 ∈ V) | |
6 | uniexg 7456 | . . . 4 ⊢ (𝑆 ∈ 𝑊 → ∪ 𝑆 ∈ V) | |
7 | xpexg 7462 | . . . 4 ⊢ ((∪ 𝑅 ∈ V ∧ ∪ 𝑆 ∈ V) → (∪ 𝑅 × ∪ 𝑆) ∈ V) | |
8 | 5, 6, 7 | syl2an 595 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∪ 𝑅 × ∪ 𝑆) ∈ V) |
9 | 4, 8 | eqeltrrid 2915 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ∪ 𝐵 ∈ V) |
10 | uniexb 7475 | . 2 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
11 | 9, 10 | sylibr 235 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∪ cuni 4830 × cxp 5546 ran crn 5549 ∈ cmpo 7147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 |
This theorem is referenced by: txbas 22103 eltx 22104 txtopon 22127 txopn 22138 txss12 22141 txbasval 22142 txrest 22167 sxsiga 31349 elsx 31352 mbfmco2 31422 |
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