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Theorem iskgen2 22249
 Description: A space is compactly generated iff it contains its image under the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
iskgen2 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))

Proof of Theorem iskgen2
StepHypRef Expression
1 kgentop 22243 . . 3 (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top)
2 kgenidm 22248 . . . 4 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽)
3 eqimss 3949 . . . 4 ((𝑘Gen‘𝐽) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽)
42, 3syl 17 . . 3 (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽)
51, 4jca 516 . 2 (𝐽 ∈ ran 𝑘Gen → (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
6 simpr 489 . . . 4 ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) ⊆ 𝐽)
7 kgenss 22244 . . . . 5 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
87adantr 485 . . . 4 ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → 𝐽 ⊆ (𝑘Gen‘𝐽))
96, 8eqssd 3910 . . 3 ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) = 𝐽)
10 kgenf 22242 . . . . . 6 𝑘Gen:Top⟶Top
11 ffn 6499 . . . . . 6 (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top)
1210, 11ax-mp 5 . . . . 5 𝑘Gen Fn Top
13 fnfvelrn 6840 . . . . 5 ((𝑘Gen Fn Top ∧ 𝐽 ∈ Top) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen)
1412, 13mpan 690 . . . 4 (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen)
1514adantr 485 . . 3 ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen)
169, 15eqeltrrd 2854 . 2 ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → 𝐽 ∈ ran 𝑘Gen)
175, 16impbii 212 1 (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ⊆ wss 3859  ran crn 5526   Fn wfn 6331  ⟶wf 6332  ‘cfv 6336  Topctop 21594  𝑘Genckgen 22234 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-en 8529  df-fin 8532  df-fi 8909  df-rest 16755  df-topgen 16776  df-top 21595  df-topon 21612  df-bases 21647  df-cmp 22088  df-kgen 22235 This theorem is referenced by:  iskgen3  22250  llycmpkgen2  22251  1stckgen  22255  txkgen  22353  qtopkgen  22411
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