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Metamath Proof Explorer

Theorem iunsuc

Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013) (Proof shortened by Mario Carneiro, 18-Nov-2016)

		Ref	Expression
	Hypotheses	iunsuc.1	$⊢ A \in V$
		iunsuc.2	$⊢ x = A \to B = C$
	Assertion	iunsuc	$⊢ ⋃_{x \in suc A} B = ⋃_{x \in A} B \cup C$

Proof

Step	Hyp	Ref	Expression
1		iunsuc.1	$⊢ A \in V$
2		iunsuc.2	$⊢ x = A \to B = C$
3		df-suc	$⊢ suc A = A \cup \{A\}$
4		iuneq1	$⊢ suc A = A \cup \{A\} \to ⋃_{x \in suc A} B = ⋃_{x \in A \cup \{A\}} B$
5	3 4	ax-mp	$⊢ ⋃_{x \in suc A} B = ⋃_{x \in A \cup \{A\}} B$
6		iunxun	$⊢ ⋃_{x \in A \cup \{A\}} B = ⋃_{x \in A} B \cup ⋃_{x \in \{A\}} B$
7	1 2	iunxsn	$⊢ ⋃_{x \in \{A\}} B = C$
8	7	uneq2i	$⊢ ⋃_{x \in A} B \cup ⋃_{x \in \{A\}} B = ⋃_{x \in A} B \cup C$
9	5 6 8	3eqtri	$⊢ ⋃_{x \in suc A} B = ⋃_{x \in A} B \cup C$