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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 8594 | . . 3 ⊢ {𝑌} ∈ Fin | |
4 | 3 | a1i 11 | . 2 ⊢ (𝜑 → {𝑌} ∈ Fin) |
5 | ptopn2.o | . . . . . 6 ⊢ (𝜑 → 𝑂 ∈ (𝐹‘𝑌)) | |
6 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑂 ∈ (𝐹‘𝑌)) |
7 | fveq2 6670 | . . . . . 6 ⊢ (𝑘 = 𝑌 → (𝐹‘𝑘) = (𝐹‘𝑌)) | |
8 | 7 | eleq2d 2898 | . . . . 5 ⊢ (𝑘 = 𝑌 → (𝑂 ∈ (𝐹‘𝑘) ↔ 𝑂 ∈ (𝐹‘𝑌))) |
9 | 6, 8 | syl5ibrcom 249 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑘 = 𝑌 → 𝑂 ∈ (𝐹‘𝑘))) |
10 | 9 | imp 409 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝑘 = 𝑌) → 𝑂 ∈ (𝐹‘𝑘)) |
11 | 2 | ffvelrnda 6851 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐹‘𝑘) ∈ Top) |
12 | eqid 2821 | . . . . . 6 ⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘) | |
13 | 12 | topopn 21514 | . . . . 5 ⊢ ((𝐹‘𝑘) ∈ Top → ∪ (𝐹‘𝑘) ∈ (𝐹‘𝑘)) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∪ (𝐹‘𝑘) ∈ (𝐹‘𝑘)) |
15 | 14 | adantr 483 | . . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝑘 = 𝑌) → ∪ (𝐹‘𝑘) ∈ (𝐹‘𝑘)) |
16 | 10, 15 | ifclda 4501 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) ∈ (𝐹‘𝑘)) |
17 | eldifn 4104 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → ¬ 𝑘 ∈ {𝑌}) | |
18 | velsn 4583 | . . . . 5 ⊢ (𝑘 ∈ {𝑌} ↔ 𝑘 = 𝑌) | |
19 | 17, 18 | sylnib 330 | . . . 4 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → ¬ 𝑘 = 𝑌) |
20 | 19 | iffalsed 4478 | . . 3 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑌}) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘)) |
21 | 20 | adantl 484 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑌})) → if(𝑘 = 𝑌, 𝑂, ∪ (𝐹‘𝑘)) = ∪ (𝐹‘𝑘)) |
22 | 1, 2, 4, 16, 21 | ptopn 22191 | 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 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ifcif 4467 {csn 4567 ∪ cuni 4838 ⟶wf 6351 ‘cfv 6355 Xcixp 8461 Fincfn 8509 ∏tcpt 16712 Topctop 21501 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-ixp 8462 df-en 8510 df-fin 8513 df-fi 8875 df-topgen 16717 df-pt 16718 df-top 21502 df-bases 21554 |
This theorem is referenced by: ptcld 22221 |
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