Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 8582 | . . 3 ⊢ {𝑌} ∈ Fin | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → {𝑌} ∈ Fin) |
5 | ptopn2.o | . . . . . 6 ⊢ (𝜑 → 𝑂 ∈ (𝐹‘𝑌)) | |
6 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑂 ∈ (𝐹‘𝑌)) |
7 | fveq2 6663 | . . . . . 6 ⊢ (𝑘 = 𝑌 → (𝐹‘𝑘) = (𝐹‘𝑌)) | |
8 | 7 | eleq2d 2895 | . . . . 5 ⊢ (𝑘 = 𝑌 → (𝑂 ∈ (𝐹‘𝑘) ↔ 𝑂 ∈ (𝐹‘𝑌))) |
9 | 6, 8 | syl5ibrcom 248 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 = 𝑌 → 𝑂 ∈ (𝐹‘𝑘))) |
10 | 9 | imp 407 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 = 𝑌) → 𝑂 ∈ (𝐹‘𝑘)) |
11 | 2 | ffvelrnda 6843 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ Top) |
12 | eqid 2818 | . . . . . 6 ⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘) | |
13 | 12 | topopn 21442 | . . . . 5 ⊢ ((𝐹‘𝑘) ∈ Top → ∪ (𝐹‘𝑘) ∈ (𝐹‘𝑘)) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∪ (𝐹‘𝑘) ∈ (𝐹‘𝑘)) |
15 | 14 | adantr 481 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 = 𝑌) → ∪ (𝐹‘𝑘) ∈ (𝐹‘𝑘)) |
16 | 10, 15 | ifclda 4497 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) ∈ (𝐹‘𝑘)) |
17 | eldifn 4101 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → ¬ 𝑘 ∈ {𝑌}) | |
18 | velsn 4573 | . . . . 5 ⊢ (𝑘 ∈ {𝑌} ↔ 𝑘 = 𝑌) | |
19 | 17, 18 | sylnib 329 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → ¬ 𝑘 = 𝑌) |
20 | 19 | iffalsed 4474 | . . 3 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘)) |
21 | 20 | adantl 482 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑌})) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘)) |
22 | 1, 2, 4, 16, 21 | ptopn 22119 | 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 396 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ifcif 4463 {csn 4557 ∪ cuni 4830 ⟶wf 6344 ‘cfv 6348 Xcixp 8449 Fincfn 8497 ∏tcpt 16700 Topctop 21429 |
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-rep 5181 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-3or 1080 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-reu 3142 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-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ixp 8450 df-en 8498 df-fin 8501 df-fi 8863 df-topgen 16705 df-pt 16706 df-top 21430 df-bases 21482 |
This theorem is referenced by: ptcld 22149 |
Copyright terms: Public domain | W3C validator |