Theorem tfr1a

Description: A weak version of tfr1 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015)

	Ref	Expression
Hypothesis	tfr.1	$$⊢F=\mathrm{recs}\left(G\right)$$
Assertion	tfr1a	$$⊢\left(\mathrm{Fun}F\wedge \mathrm{Lim}\mathrm{dom}F\right)$$

Proof

Step	Hyp	Ref	Expression
1		tfr.1	$$⊢F=\mathrm{recs}\left(G\right)$$
2		eqid	$$⊢\left\{f|\exists x\in \mathrm{On}\left(fFnx\wedge \forall y\in xf\left(y\right)=G\left({f\upharpoonright }_y\right)\right)\right\}=\left\{f|\exists x\in \mathrm{On}\left(fFnx\wedge \forall y\in xf\left(y\right)=G\left({f\upharpoonright }_y\right)\right)\right\}$$
3	2	tfrlem7	$$⊢\mathrm{Fun}\mathrm{recs}\left(G\right)$$
4	1	funeqi	$$⊢\mathrm{Fun}F\leftrightarrow \mathrm{Fun}\mathrm{recs}\left(G\right)$$
5	3 4	mpbir	$$⊢\mathrm{Fun}F$$
6	2	tfrlem16	$$⊢\mathrm{Lim}\mathrm{dom}\mathrm{recs}\left(G\right)$$
7	1	dmeqi	$$⊢\mathrm{dom}F=\mathrm{dom}\mathrm{recs}\left(G\right)$$
8		limeq	$$⊢\mathrm{dom}F=\mathrm{dom}\mathrm{recs}\left(G\right)\to \left(\mathrm{Lim}\mathrm{dom}F\leftrightarrow \mathrm{Lim}\mathrm{dom}\mathrm{recs}\left(G\right)\right)$$
9	7 8	ax-mp	$$⊢\mathrm{Lim}\mathrm{dom}F\leftrightarrow \mathrm{Lim}\mathrm{dom}\mathrm{recs}\left(G\right)$$
10	6 9	mpbir	$$⊢\mathrm{Lim}\mathrm{dom}F$$
11	5 10	pm3.2i	$$⊢\left(\mathrm{Fun}F\wedge \mathrm{Lim}\mathrm{dom}F\right)$$