This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem infmap

Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of Jech p. 43. (Contributed by NM, 1-Oct-2004) (Proof shortened by Mario Carneiro, 30-Apr-2015)

		Ref	Expression
	Assertion	infmap	$⊢ (ω ≼ A \land B ≼ A) \to (A^{B}) \approx \{x \| (x \subseteq A \land x \approx B)\}$

Proof

Step	Hyp	Ref	Expression
1		ovex	$⊢ A^{B} \in V$
2		numth3	$⊢ A^{B} \in V \to A^{B} \in dom card$
3	1 2	ax-mp	$⊢ A^{B} \in dom card$
4		infmap2	$⊢ (ω ≼ A \land B ≼ A \land A^{B} \in dom card) \to (A^{B}) \approx \{x \| (x \subseteq A \land x \approx B)\}$
5	3 4	mp3an3	$⊢ (ω ≼ A \land B ≼ A) \to (A^{B}) \approx \{x \| (x \subseteq A \land x \approx B)\}$