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Mirrors > Home > MPE Home > Th. List > tgiun | Structured version Visualization version GIF version |
Description: The indexed union of a set of basic open sets is in the generated topology. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tgiun | ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ (topGen‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfiun3g 5835 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
2 | 1 | adantl 484 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶)) |
3 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
4 | 3 | rnmptss 6886 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 → ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ 𝐵) |
5 | eltg3i 21569 | . . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ 𝐵) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (topGen‘𝐵)) | |
6 | 4, 5 | sylan2 594 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (topGen‘𝐵)) |
7 | 2, 6 | eqeltrd 2913 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵) → ∪ 𝑥 ∈ 𝐴 𝐶 ∈ (topGen‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ⊆ wss 3936 ∪ cuni 4838 ∪ ciun 4919 ↦ cmpt 5146 ran crn 5556 ‘cfv 6355 topGenctg 16711 |
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-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-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-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 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-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-topgen 16717 |
This theorem is referenced by: txbasval 22214 |
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