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Metamath Proof Explorer

Theorem cardf

Description: The cardinality function is a function with domain the well-orderable sets. Assuming AC, this is the universe. (Contributed by Mario Carneiro, 6-Jun-2013) (Revised by Mario Carneiro, 13-Sep-2013)

		Ref	Expression
	Assertion	cardf	$⊢ card : V ⟶ On$

Proof

Step	Hyp	Ref	Expression
1		cardf2	$⊢ card : \{x \| \exists y \in On y \approx x\} ⟶ On$
2	1	fdmi	$⊢ dom card = \{x \| \exists y \in On y \approx x\}$
3		cardeqv	$⊢ dom card = V$
4	2 3	eqtr3i	$⊢ \{x \| \exists y \in On y \approx x\} = V$
5	4	feq2i	$⊢ card : \{x \| \exists y \in On y \approx x\} ⟶ On \leftrightarrow card : V ⟶ On$
6	1 5	mpbi	$⊢ card : V ⟶ On$