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Mirrors > Home > MPE Home > Th. List > ptbasin2 | Structured version Visualization version GIF version |
Description: The basis for a product topology is closed under intersections. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
ptbas.1 | ⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} |
Ref | Expression |
---|---|
ptbasin2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ptbas.1 | . . . 4 ⊢ 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} | |
2 | 1 | ptbasin 23474 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢 ∩ 𝑣) ∈ 𝐵) |
3 | 2 | ralrimivva 3195 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ∩ 𝑣) ∈ 𝐵) |
4 | 1 | ptuni2 23473 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) = ∪ 𝐵) |
5 | ixpexg 8934 | . . . . . 6 ⊢ (∀𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V → X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V) | |
6 | fvex 6904 | . . . . . . . 8 ⊢ (𝐹‘𝑘) ∈ V | |
7 | 6 | uniex 7740 | . . . . . . 7 ⊢ ∪ (𝐹‘𝑘) ∈ V |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ∪ (𝐹‘𝑘) ∈ V) |
9 | 5, 8 | mprg 3062 | . . . . 5 ⊢ X𝑘 ∈ 𝐴 ∪ (𝐹‘𝑘) ∈ V |
10 | 4, 9 | eqeltrrdi 2837 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ∪ 𝐵 ∈ V) |
11 | uniexb 7760 | . . . 4 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
12 | 10, 11 | sylibr 233 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐵 ∈ V) |
13 | inficl 9442 | . . 3 ⊢ (𝐵 ∈ V → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ∩ 𝑣) ∈ 𝐵 ↔ (fi‘𝐵) = 𝐵)) | |
14 | 12, 13 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 (𝑢 ∩ 𝑣) ∈ 𝐵 ↔ (fi‘𝐵) = 𝐵)) |
15 | 3, 14 | mpbid 231 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘𝐵) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 {cab 2704 ∀wral 3056 ∃wrex 3065 Vcvv 3469 ∖ cdif 3941 ∩ cin 3943 ∪ cuni 4903 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 Xcixp 8909 Fincfn 8957 ficfi 9427 Topctop 22788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-om 7865 df-1o 8480 df-er 8718 df-ixp 8910 df-en 8958 df-fin 8961 df-fi 9428 df-top 22789 |
This theorem is referenced by: ptbas 23476 ptbasfi 23478 |
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