Theorem cfval

Description: Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number A is the cardinality (size) of the smallest unbounded subset y of the ordinal number. Unbounded means that for every member of A , there is a member of y that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004) (Revised by Mario Carneiro, 15-Sep-2013)

	Ref	Expression
Assertion	cfval	⊢ ( 𝐴 ∈ On → ( cf ‘ 𝐴 ) = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } )

Proof

Step	Hyp	Ref	Expression
1		cflem	⊢ ( 𝐴 ∈ On → ∃ 𝑥 ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) )
2		intexab	⊢ ( ∃ 𝑥 ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ∈ V )
3	1 2	sylib	⊢ ( 𝐴 ∈ On → ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ∈ V )
4		sseq2	⊢ ( 𝑣 = 𝐴 → ( 𝑦 ⊆ 𝑣 ↔ 𝑦 ⊆ 𝐴 ) )
5		raleq	⊢ ( 𝑣 = 𝐴 → ( ∀ 𝑧 ∈ 𝑣 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) )
6	4 5	anbi12d	⊢ ( 𝑣 = 𝐴 → ( ( 𝑦 ⊆ 𝑣 ∧ ∀ 𝑧 ∈ 𝑣 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ↔ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) )
7	6	anbi2d	⊢ ( 𝑣 = 𝐴 → ( ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝑣 ∧ ∀ 𝑧 ∈ 𝑣 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) )
8	7	exbidv	⊢ ( 𝑣 = 𝐴 → ( ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝑣 ∧ ∀ 𝑧 ∈ 𝑣 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ↔ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) ) )
9	8	abbidv	⊢ ( 𝑣 = 𝐴 → { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝑣 ∧ ∀ 𝑧 ∈ 𝑣 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } = { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } )
10	9	inteqd	⊢ ( 𝑣 = 𝐴 → ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝑣 ∧ ∀ 𝑧 ∈ 𝑣 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } )
11		df-cf	⊢ cf = ( 𝑣 ∈ On ↦ ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝑣 ∧ ∀ 𝑧 ∈ 𝑣 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } )
12	10 11	fvmptg	⊢ ( ( 𝐴 ∈ On ∧ ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } ∈ V ) → ( cf ‘ 𝐴 ) = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } )
13	3 12	mpdan	⊢ ( 𝐴 ∈ On → ( cf ‘ 𝐴 ) = ∩ { 𝑥 ∣ ∃ 𝑦 ( 𝑥 = ( card ‘ 𝑦 ) ∧ ( 𝑦 ⊆ 𝐴 ∧ ∀ 𝑧 ∈ 𝐴 ∃ 𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ) ) } )