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Mirrors > Home > MPE Home > Th. List > tgfiss | Structured version Visualization version GIF version |
Description: If a subbase is included into a topology, so is the generated topology. (Contributed by FL, 20-Apr-2012.) (Proof shortened by Mario Carneiro, 10-Jan-2015.) |
Ref | Expression |
---|---|
tgfiss | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (topGen‘(fi‘𝐴)) ⊆ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fiss 8882 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (fi‘𝐴) ⊆ (fi‘𝐽)) | |
2 | fitop 21502 | . . . . 5 ⊢ (𝐽 ∈ Top → (fi‘𝐽) = 𝐽) | |
3 | 2 | adantr 483 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (fi‘𝐽) = 𝐽) |
4 | 1, 3 | sseqtrd 4006 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (fi‘𝐴) ⊆ 𝐽) |
5 | tgss 21570 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (fi‘𝐴) ⊆ 𝐽) → (topGen‘(fi‘𝐴)) ⊆ (topGen‘𝐽)) | |
6 | 4, 5 | syldan 593 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (topGen‘(fi‘𝐴)) ⊆ (topGen‘𝐽)) |
7 | tgtop 21575 | . . 3 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽) | |
8 | 7 | adantr 483 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (topGen‘𝐽) = 𝐽) |
9 | 6, 8 | sseqtrd 4006 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐽) → (topGen‘(fi‘𝐴)) ⊆ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 ficfi 8868 topGenctg 16705 Topctop 21495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-fin 8507 df-fi 8869 df-topgen 16711 df-top 21496 |
This theorem is referenced by: ordtrest 21804 ordtrest2 21806 lecldbas 21821 xkoptsub 22256 xkopt 22257 ordtrestNEW 31159 ordtrest2NEW 31161 topjoin 33708 |
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