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Metamath Proof Explorer

Theorem qtopcmplem

Description: Lemma for qtopcmp and qtopconn . (Contributed by Mario Carneiro, 24-Mar-2015)

Ref Expression
Hypotheses qtopcmp.1 ⊢ 𝑋 = ∪ 𝐽
qtopcmplem.1 ⊢ ( 𝐽 ∈ 𝐴 → 𝐽 ∈ Top )
qtopcmplem.2 ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 : 𝑋 –onto→ ∪ ( 𝐽 qTop 𝐹 ) ∧ 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 )
Assertion qtopcmplem ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 qtopcmp.1 ⊢ 𝑋 = ∪ 𝐽
2 qtopcmplem.1 ⊢ ( 𝐽 ∈ 𝐴 → 𝐽 ∈ Top )
3 qtopcmplem.2 ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 : 𝑋 –onto→ ∪ ( 𝐽 qTop 𝐹 ) ∧ 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 )
4 simpl ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐽 ∈ 𝐴 )
5 simpr ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐹 Fn 𝑋 )
6 dffn4 ⊢ ( 𝐹 Fn 𝑋 ↔ 𝐹 : 𝑋 –onto→ ran 𝐹 )
7 5 6 sylib ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐹 : 𝑋 –onto→ ran 𝐹 )
8 1 qtopuni ⊢ ( ( 𝐽 ∈ Top ∧ 𝐹 : 𝑋 –onto→ ran 𝐹 ) → ran 𝐹 = ∪ ( 𝐽 qTop 𝐹 ) )
9 2 8 sylan ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 : 𝑋 –onto→ ran 𝐹 ) → ran 𝐹 = ∪ ( 𝐽 qTop 𝐹 ) )
10 6 9 sylan2b ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → ran 𝐹 = ∪ ( 𝐽 qTop 𝐹 ) )
11 foeq3 ⊢ ( ran 𝐹 = ∪ ( 𝐽 qTop 𝐹 ) → ( 𝐹 : 𝑋 –onto→ ran 𝐹 ↔ 𝐹 : 𝑋 –onto→ ∪ ( 𝐽 qTop 𝐹 ) ) )
12 10 11 syl ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → ( 𝐹 : 𝑋 –onto→ ran 𝐹 ↔ 𝐹 : 𝑋 –onto→ ∪ ( 𝐽 qTop 𝐹 ) ) )
13 7 12 mpbid ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐹 : 𝑋 –onto→ ∪ ( 𝐽 qTop 𝐹 ) )
14 1 toptopon ⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
15 2 14 sylib ⊢ ( 𝐽 ∈ 𝐴 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
16 qtopid ⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 Fn 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) )
17 15 16 sylan ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → 𝐹 ∈ ( 𝐽 Cn ( 𝐽 qTop 𝐹 ) ) )
18 4 13 17 3 syl3anc ⊢ ( ( 𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋 ) → ( 𝐽 qTop 𝐹 ) ∈ 𝐴 )