wfrfun

Description: The "function" generated by the well-ordered recursion generator is indeed a function. Avoids the axiom of replacement. (Contributed by Scott Fenton, 21-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015) (Revised by Scott Fenton, 17-Nov-2024)

		Ref	Expression
	Hypothesis	wfrfun.1	$⊢ F = wrecs (R, A, G)$
	Assertion	wfrfun	$⊢ (R We A \land R Se A) \to Fun F$

Proof

Step	Hyp	Ref	Expression
1		wfrfun.1	$⊢ F = wrecs (R, A, G)$
2		wefr	$⊢ R We A \to R Fr A$
3	2	adantr	$⊢ (R We A \land R Se A) \to R Fr A$
4		weso	$⊢ R We A \to R Or A$
5		sopo	$⊢ R Or A \to R Po A$
6	4 5	syl	$⊢ R We A \to R Po A$
7	6	adantr	$⊢ (R We A \land R Se A) \to R Po A$
8		simpr	$⊢ (R We A \land R Se A) \to R Se A$
9		df-wrecs	$⊢ wrecs (R, A, G) = frecs (R, A, G \circ 2^{nd})$
10	1 9	eqtri	$⊢ F = frecs (R, A, G \circ 2^{nd})$
11	10	fprfung	$⊢ (R Fr A \land R Po A \land R Se A) \to Fun F$
12	3 7 8 11	syl3anc	$⊢ (R We A \land R Se A) \to Fun F$

Metamath Proof Explorer

Theorem wfrfun

Proof