Theorem tskurn

Description: A transitive Tarski class is closed under small unions. (Contributed by Mario Carneiro, 22-Jun-2013)

	Ref	Expression
Assertion	tskurn	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → ∪ ran 𝐹 ∈ 𝑇 )

Proof

Step	Hyp	Ref	Expression
1		simp1l	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → 𝑇 ∈ Tarski )
2		simp1r	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → Tr 𝑇 )
3		frn	⊢ ( 𝐹 : 𝐴 ⟶ 𝑇 → ran 𝐹 ⊆ 𝑇 )
4	3	3ad2ant3	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → ran 𝐹 ⊆ 𝑇 )
5		tskwe2	⊢ ( 𝑇 ∈ Tarski → 𝑇 ∈ dom card )
6	1 5	syl	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → 𝑇 ∈ dom card )
7		simp2	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → 𝐴 ∈ 𝑇 )
8		trss	⊢ ( Tr 𝑇 → ( 𝐴 ∈ 𝑇 → 𝐴 ⊆ 𝑇 ) )
9	2 7 8	sylc	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → 𝐴 ⊆ 𝑇 )
10		ssnum	⊢ ( ( 𝑇 ∈ dom card ∧ 𝐴 ⊆ 𝑇 ) → 𝐴 ∈ dom card )
11	6 9 10	syl2anc	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → 𝐴 ∈ dom card )
12		ffn	⊢ ( 𝐹 : 𝐴 ⟶ 𝑇 → 𝐹 Fn 𝐴 )
13		dffn4	⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 : 𝐴 –onto→ ran 𝐹 )
14	12 13	sylib	⊢ ( 𝐹 : 𝐴 ⟶ 𝑇 → 𝐹 : 𝐴 –onto→ ran 𝐹 )
15	14	3ad2ant3	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → 𝐹 : 𝐴 –onto→ ran 𝐹 )
16		fodomnum	⊢ ( 𝐴 ∈ dom card → ( 𝐹 : 𝐴 –onto→ ran 𝐹 → ran 𝐹 ≼ 𝐴 ) )
17	11 15 16	sylc	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → ran 𝐹 ≼ 𝐴 )
18		tsksdom	⊢ ( ( 𝑇 ∈ Tarski ∧ 𝐴 ∈ 𝑇 ) → 𝐴 ≺ 𝑇 )
19	1 7 18	syl2anc	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → 𝐴 ≺ 𝑇 )
20		domsdomtr	⊢ ( ( ran 𝐹 ≼ 𝐴 ∧ 𝐴 ≺ 𝑇 ) → ran 𝐹 ≺ 𝑇 )
21	17 19 20	syl2anc	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → ran 𝐹 ≺ 𝑇 )
22		tskssel	⊢ ( ( 𝑇 ∈ Tarski ∧ ran 𝐹 ⊆ 𝑇 ∧ ran 𝐹 ≺ 𝑇 ) → ran 𝐹 ∈ 𝑇 )
23	1 4 21 22	syl3anc	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → ran 𝐹 ∈ 𝑇 )
24		tskuni	⊢ ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ∧ ran 𝐹 ∈ 𝑇 ) → ∪ ran 𝐹 ∈ 𝑇 )
25	1 2 23 24	syl3anc	⊢ ( ( ( 𝑇 ∈ Tarski ∧ Tr 𝑇 ) ∧ 𝐴 ∈ 𝑇 ∧ 𝐹 : 𝐴 ⟶ 𝑇 ) → ∪ ran 𝐹 ∈ 𝑇 )