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Metamath Proof Explorer

Theorem carduniima

Description: The union of the image of a mapping to cardinals is a cardinal. Proposition 11.16 of TakeutiZaring p. 104. (Contributed by NM, 4-Nov-2004)

		Ref	Expression
	Assertion	carduniima	$⊢ A \in B \to (F : A ⟶ ω \cup ran ℵ \to ⋃ F [A] \in (ω \cup ran ℵ))$

Proof

Step	Hyp	Ref	Expression
1		ffun	$⊢ F : A ⟶ ω \cup ran ℵ \to Fun F$
2		funimaexg	$⊢ (Fun F \land A \in B) \to F [A] \in V$
3	1 2	sylan	$⊢ (F : A ⟶ ω \cup ran ℵ \land A \in B) \to F [A] \in V$
4	3	expcom	$⊢ A \in B \to (F : A ⟶ ω \cup ran ℵ \to F [A] \in V)$
5		fimass	$⊢ F : A ⟶ ω \cup ran ℵ \to F [A] \subseteq ω \cup ran ℵ$
6	5	sseld	$⊢ F : A ⟶ ω \cup ran ℵ \to (x \in F [A] \to x \in (ω \cup ran ℵ))$
7		iscard3	$⊢ card (x) = x \leftrightarrow x \in (ω \cup ran ℵ)$
8	6 7	imbitrrdi	$⊢ F : A ⟶ ω \cup ran ℵ \to (x \in F [A] \to card (x) = x)$
9	8	ralrimiv	$⊢ F : A ⟶ ω \cup ran ℵ \to \forall x \in F [A] card (x) = x$
10		carduni	$⊢ F [A] \in V \to (\forall x \in F [A] card (x) = x \to card (⋃ F [A]) = ⋃ F [A])$
11	9 10	syl5	$⊢ F [A] \in V \to (F : A ⟶ ω \cup ran ℵ \to card (⋃ F [A]) = ⋃ F [A])$
12	4 11	syli	$⊢ A \in B \to (F : A ⟶ ω \cup ran ℵ \to card (⋃ F [A]) = ⋃ F [A])$
13		iscard3	$⊢ card (⋃ F [A]) = ⋃ F [A] \leftrightarrow ⋃ F [A] \in (ω \cup ran ℵ)$
14	12 13	imbitrdi	$⊢ A \in B \to (F : A ⟶ ω \cup ran ℵ \to ⋃ F [A] \in (ω \cup ran ℵ))$