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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 2823 | . . . 4 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
3 | eqid 2823 | . . . 4 ⊢ ∪ 𝑆 = ∪ 𝑆 | |
4 | 1, 2, 3 | txuni2 22175 | . . 3 ⊢ (∪ 𝑅 × ∪ 𝑆) = ∪ 𝐵 |
5 | uniexg 7468 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → ∪ 𝑅 ∈ V) | |
6 | uniexg 7468 | . . . 4 ⊢ (𝑆 ∈ 𝑊 → ∪ 𝑆 ∈ V) | |
7 | xpexg 7475 | . . . 4 ⊢ ((∪ 𝑅 ∈ V ∧ ∪ 𝑆 ∈ V) → (∪ 𝑅 × ∪ 𝑆) ∈ V) | |
8 | 5, 6, 7 | syl2an 597 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (∪ 𝑅 × ∪ 𝑆) ∈ V) |
9 | 4, 8 | eqeltrrid 2920 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ∪ 𝐵 ∈ V) |
10 | uniexb 7488 | . 2 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
11 | 9, 10 | sylibr 236 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cuni 4840 × cxp 5555 ran crn 5558 ∈ cmpo 7160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 |
This theorem is referenced by: txbas 22177 eltx 22178 txtopon 22201 txopn 22212 txss12 22215 txbasval 22216 txrest 22241 sxsiga 31452 elsx 31455 mbfmco2 31525 |
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