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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 8038 | . . 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 6191 | . . . . . 6 ⊢ (𝑘 = 𝑌 → (𝐹‘𝑘) = (𝐹‘𝑌)) | |
8 | 7 | eleq2d 2687 | . . . . 5 ⊢ (𝑘 = 𝑌 → (𝑂 ∈ (𝐹‘𝑘) ↔ 𝑂 ∈ (𝐹‘𝑌))) |
9 | 6, 8 | syl5ibrcom 237 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 = 𝑌 → 𝑂 ∈ (𝐹‘𝑘))) |
10 | 9 | imp 445 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 = 𝑌) → 𝑂 ∈ (𝐹‘𝑘)) |
11 | 2 | ffvelrnda 6359 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ Top) |
12 | eqid 2622 | . . . . . 6 ⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘) | |
13 | 12 | topopn 20711 | . . . . 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 4120 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) ∈ (𝐹‘𝑘)) |
17 | eldifn 3733 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → ¬ 𝑘 ∈ {𝑌}) | |
18 | velsn 4193 | . . . . 5 ⊢ (𝑘 ∈ {𝑌} ↔ 𝑘 = 𝑌) | |
19 | 17, 18 | sylnib 318 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → ¬ 𝑘 = 𝑌) |
20 | 19 | iffalsed 4097 | . . 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 21386 | 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 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ifcif 4086 {csn 4177 ∪ cuni 4436 ⟶wf 5884 ‘cfv 5888 Xcixp 7908 Fincfn 7955 ∏tcpt 16099 Topctop 20698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-ixp 7909 df-en 7956 df-fin 7959 df-fi 8317 df-topgen 16104 df-pt 16105 df-top 20699 df-bases 20750 |
This theorem is referenced by: ptcld 21416 |
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