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Mirrors > Home > MPE Home > Th. List > ptopn2 | Structured version Visualization version GIF version |
Description: A sub-basic open set in the product topology. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
Ref | Expression |
---|---|
ptopn2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ptopn2.f | ⊢ (𝜑 → 𝐹:𝐴⟶Top) |
ptopn2.o | ⊢ (𝜑 → 𝑂 ∈ (𝐹‘𝑌)) |
Ref | Expression |
---|---|
ptopn2 | ⊢ (𝜑 → X𝑘 ∈ 𝐴 if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) ∈ (∏t‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ptopn2.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | ptopn2.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶Top) | |
3 | snfi 9082 | . . 3 ⊢ {𝑌} ∈ Fin | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → {𝑌} ∈ Fin) |
5 | ptopn2.o | . . . . . 6 ⊢ (𝜑 → 𝑂 ∈ (𝐹‘𝑌)) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑂 ∈ (𝐹‘𝑌)) |
7 | fveq2 6907 | . . . . . 6 ⊢ (𝑘 = 𝑌 → (𝐹‘𝑘) = (𝐹‘𝑌)) | |
8 | 7 | eleq2d 2825 | . . . . 5 ⊢ (𝑘 = 𝑌 → (𝑂 ∈ (𝐹‘𝑘) ↔ 𝑂 ∈ (𝐹‘𝑌))) |
9 | 6, 8 | syl5ibrcom 247 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 = 𝑌 → 𝑂 ∈ (𝐹‘𝑘))) |
10 | 9 | imp 406 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 = 𝑌) → 𝑂 ∈ (𝐹‘𝑘)) |
11 | 2 | ffvelcdmda 7104 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ Top) |
12 | eqid 2735 | . . . . . 6 ⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘) | |
13 | 12 | topopn 22928 | . . . . 5 ⊢ ((𝐹‘𝑘) ∈ Top → ∪ (𝐹‘𝑘) ∈ (𝐹‘𝑘)) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∪ (𝐹‘𝑘) ∈ (𝐹‘𝑘)) |
15 | 14 | adantr 480 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 = 𝑌) → ∪ (𝐹‘𝑘) ∈ (𝐹‘𝑘)) |
16 | 10, 15 | ifclda 4566 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) ∈ (𝐹‘𝑘)) |
17 | eldifn 4142 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → ¬ 𝑘 ∈ {𝑌}) | |
18 | velsn 4647 | . . . . 5 ⊢ (𝑘 ∈ {𝑌} ↔ 𝑘 = 𝑌) | |
19 | 17, 18 | sylnib 328 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → ¬ 𝑘 = 𝑌) |
20 | 19 | iffalsed 4542 | . . 3 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘)) |
21 | 20 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑌})) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘)) |
22 | 1, 2, 4, 16, 21 | ptopn 23607 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝐴 if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) ∈ (∏t‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ifcif 4531 {csn 4631 ∪ cuni 4912 ⟶wf 6559 ‘cfv 6563 Xcixp 8936 Fincfn 8984 ∏tcpt 17485 Topctop 22915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-2o 8506 df-ixp 8937 df-en 8985 df-fin 8988 df-fi 9449 df-topgen 17490 df-pt 17491 df-top 22916 df-bases 22969 |
This theorem is referenced by: ptcld 23637 |
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