Metamath Proof Explorer


Theorem imastopn

Description: The topology of an image structure. (Contributed by Mario Carneiro, 27-Aug-2015)

Ref Expression
Hypotheses imastps.u ( 𝜑𝑈 = ( 𝐹s 𝑅 ) )
imastps.v ( 𝜑𝑉 = ( Base ‘ 𝑅 ) )
imastps.f ( 𝜑𝐹 : 𝑉onto𝐵 )
imastopn.r ( 𝜑𝑅𝑊 )
imastopn.j 𝐽 = ( TopOpen ‘ 𝑅 )
imastopn.o 𝑂 = ( TopOpen ‘ 𝑈 )
Assertion imastopn ( 𝜑𝑂 = ( 𝐽 qTop 𝐹 ) )

Proof

Step Hyp Ref Expression
1 imastps.u ( 𝜑𝑈 = ( 𝐹s 𝑅 ) )
2 imastps.v ( 𝜑𝑉 = ( Base ‘ 𝑅 ) )
3 imastps.f ( 𝜑𝐹 : 𝑉onto𝐵 )
4 imastopn.r ( 𝜑𝑅𝑊 )
5 imastopn.j 𝐽 = ( TopOpen ‘ 𝑅 )
6 imastopn.o 𝑂 = ( TopOpen ‘ 𝑈 )
7 eqid ( TopSet ‘ 𝑈 ) = ( TopSet ‘ 𝑈 )
8 1 2 3 4 5 7 imastset ( 𝜑 → ( TopSet ‘ 𝑈 ) = ( 𝐽 qTop 𝐹 ) )
9 5 fvexi 𝐽 ∈ V
10 fofn ( 𝐹 : 𝑉onto𝐵𝐹 Fn 𝑉 )
11 3 10 syl ( 𝜑𝐹 Fn 𝑉 )
12 fvex ( Base ‘ 𝑅 ) ∈ V
13 2 12 eqeltrdi ( 𝜑𝑉 ∈ V )
14 fnex ( ( 𝐹 Fn 𝑉𝑉 ∈ V ) → 𝐹 ∈ V )
15 11 13 14 syl2anc ( 𝜑𝐹 ∈ V )
16 eqid 𝐽 = 𝐽
17 16 qtopval ( ( 𝐽 ∈ V ∧ 𝐹 ∈ V ) → ( 𝐽 qTop 𝐹 ) = { 𝑥 ∈ 𝒫 ( 𝐹 𝐽 ) ∣ ( ( 𝐹𝑥 ) ∩ 𝐽 ) ∈ 𝐽 } )
18 9 15 17 sylancr ( 𝜑 → ( 𝐽 qTop 𝐹 ) = { 𝑥 ∈ 𝒫 ( 𝐹 𝐽 ) ∣ ( ( 𝐹𝑥 ) ∩ 𝐽 ) ∈ 𝐽 } )
19 8 18 eqtrd ( 𝜑 → ( TopSet ‘ 𝑈 ) = { 𝑥 ∈ 𝒫 ( 𝐹 𝐽 ) ∣ ( ( 𝐹𝑥 ) ∩ 𝐽 ) ∈ 𝐽 } )
20 ssrab2 { 𝑥 ∈ 𝒫 ( 𝐹 𝐽 ) ∣ ( ( 𝐹𝑥 ) ∩ 𝐽 ) ∈ 𝐽 } ⊆ 𝒫 ( 𝐹 𝐽 )
21 imassrn ( 𝐹 𝐽 ) ⊆ ran 𝐹
22 forn ( 𝐹 : 𝑉onto𝐵 → ran 𝐹 = 𝐵 )
23 3 22 syl ( 𝜑 → ran 𝐹 = 𝐵 )
24 1 2 3 4 imasbas ( 𝜑𝐵 = ( Base ‘ 𝑈 ) )
25 23 24 eqtrd ( 𝜑 → ran 𝐹 = ( Base ‘ 𝑈 ) )
26 21 25 sseqtrid ( 𝜑 → ( 𝐹 𝐽 ) ⊆ ( Base ‘ 𝑈 ) )
27 26 sspwd ( 𝜑 → 𝒫 ( 𝐹 𝐽 ) ⊆ 𝒫 ( Base ‘ 𝑈 ) )
28 20 27 sstrid ( 𝜑 → { 𝑥 ∈ 𝒫 ( 𝐹 𝐽 ) ∣ ( ( 𝐹𝑥 ) ∩ 𝐽 ) ∈ 𝐽 } ⊆ 𝒫 ( Base ‘ 𝑈 ) )
29 19 28 eqsstrd ( 𝜑 → ( TopSet ‘ 𝑈 ) ⊆ 𝒫 ( Base ‘ 𝑈 ) )
30 eqid ( Base ‘ 𝑈 ) = ( Base ‘ 𝑈 )
31 30 7 topnid ( ( TopSet ‘ 𝑈 ) ⊆ 𝒫 ( Base ‘ 𝑈 ) → ( TopSet ‘ 𝑈 ) = ( TopOpen ‘ 𝑈 ) )
32 29 31 syl ( 𝜑 → ( TopSet ‘ 𝑈 ) = ( TopOpen ‘ 𝑈 ) )
33 32 6 eqtr4di ( 𝜑 → ( TopSet ‘ 𝑈 ) = 𝑂 )
34 33 8 eqtr3d ( 𝜑𝑂 = ( 𝐽 qTop 𝐹 ) )