grucollcld

Description: A Grothendieck universe contains the output of a collection operation whenever its left input is a relation on the universe, and its right input is in the universe. (Contributed by Rohan Ridenour, 11-Aug-2023)

	Ref	Expression
Hypotheses	grucollcld.1	$⊢ φ \to G \in Univ$
	grucollcld.2	$⊢ φ \to F \subseteq G \times G$
	grucollcld.3	$⊢ φ \to A \in G$
Assertion	grucollcld	$⊢ φ \to (F Coll A) \in G$

Proof

Step	Hyp	Ref	Expression
1		grucollcld.1	$⊢ φ \to G \in Univ$
2		grucollcld.2	$⊢ φ \to F \subseteq G \times G$
3		grucollcld.3	$⊢ φ \to A \in G$
4		dfcoll2	$⊢ (F Coll A) = ⋃_{x \in A} Scott \{y \| x F y\}$
5		simpr	$⊢ ((φ \land x \in A) \land Scott \{y \| x F y\} = \emptyset) \to Scott \{y \| x F y\} = \emptyset$
6	1	ad2antrr	$⊢ ((φ \land x \in A) \land Scott \{y \| x F y\} = \emptyset) \to G \in Univ$
7	3	ad2antrr	$⊢ ((φ \land x \in A) \land Scott \{y \| x F y\} = \emptyset) \to A \in G$
8	6 7	gru0eld	$⊢ ((φ \land x \in A) \land Scott \{y \| x F y\} = \emptyset) \to \emptyset \in G$
9	5 8	eqeltrd	$⊢ ((φ \land x \in A) \land Scott \{y \| x F y\} = \emptyset) \to Scott \{y \| x F y\} \in G$
10		neq0	$⊢ \neg Scott \{y \| x F y\} = \emptyset \leftrightarrow \exists z z \in Scott \{y \| x F y\}$
11	1	ad2antrr	$⊢ ((φ \land x \in A) \land z \in Scott \{y \| x F y\}) \to G \in Univ$
12		breq2	$⊢ y = z \to (x F y \leftrightarrow x F z)$
13	12	elscottab	$⊢ z \in Scott \{y \| x F y\} \to x F z$
14	13	adantl	$⊢ ((φ \land x \in A) \land z \in Scott \{y \| x F y\}) \to x F z$
15	2	ad2antrr	$⊢ ((φ \land x \in A) \land z \in Scott \{y \| x F y\}) \to F \subseteq G \times G$
16	15	ssbrd	$⊢ ((φ \land x \in A) \land z \in Scott \{y \| x F y\}) \to (x F z \to x (G \times G) z)$
17	14 16	mpd	$⊢ ((φ \land x \in A) \land z \in Scott \{y \| x F y\}) \to x (G \times G) z$
18		brxp	$⊢ x (G \times G) z \leftrightarrow (x \in G \land z \in G)$
19	18	simprbi	$⊢ x (G \times G) z \to z \in G$
20	17 19	syl	$⊢ ((φ \land x \in A) \land z \in Scott \{y \| x F y\}) \to z \in G$
21		simpr	$⊢ ((φ \land x \in A) \land z \in Scott \{y \| x F y\}) \to z \in Scott \{y \| x F y\}$
22	11 20 21	gruscottcld	$⊢ ((φ \land x \in A) \land z \in Scott \{y \| x F y\}) \to Scott \{y \| x F y\} \in G$
23	22	expcom	$⊢ z \in Scott \{y \| x F y\} \to ((φ \land x \in A) \to Scott \{y \| x F y\} \in G)$
24	23	exlimiv	$⊢ \exists z z \in Scott \{y \| x F y\} \to ((φ \land x \in A) \to Scott \{y \| x F y\} \in G)$
25	10 24	sylbi	$⊢ \neg Scott \{y \| x F y\} = \emptyset \to ((φ \land x \in A) \to Scott \{y \| x F y\} \in G)$
26	25	impcom	$⊢ ((φ \land x \in A) \land \neg Scott \{y \| x F y\} = \emptyset) \to Scott \{y \| x F y\} \in G$
27	9 26	pm2.61dan	$⊢ (φ \land x \in A) \to Scott \{y \| x F y\} \in G$
28	27	ralrimiva	$⊢ φ \to \forall x \in A Scott \{y \| x F y\} \in G$
29		gruiun	$⊢ (G \in Univ \land A \in G \land \forall x \in A Scott \{y \| x F y\} \in G) \to ⋃_{x \in A} Scott \{y \| x F y\} \in G$
30	1 3 28 29	syl3anc	$⊢ φ \to ⋃_{x \in A} Scott \{y \| x F y\} \in G$
31	4 30	eqeltrid	$⊢ φ \to (F Coll A) \in G$

Metamath Proof Explorer

Theorem grucollcld

Proof