Metamath Proof Explorer

Theorem wfrdmssOLD

Description: Obsolete proof of wfrdmss as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011)

		Ref	Expression
	Hypothesis	wfrlem6OLD.1	$⊢ F = wrecs (R, A, G)$
	Assertion	wfrdmssOLD	$⊢ dom F \subseteq A$

Proof

Step	Hyp	Ref	Expression
1		wfrlem6OLD.1	$⊢ F = wrecs (R, A, G)$
2		dfwrecsOLD	$⊢ wrecs (R, A, G) = ⋃ \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\}$
3	1 2	eqtri	$⊢ F = ⋃ \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\}$
4	3	dmeqi	$⊢ dom F = dom ⋃ \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\}$
5		dmuni	$⊢ dom ⋃ \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\} = ⋃_{g \in \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\}} dom g$
6	4 5	eqtri	$⊢ dom F = ⋃_{g \in \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\}} dom g$
7		iunss	$⊢ ⋃_{g \in \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\}} dom g \subseteq A \leftrightarrow \forall g \in \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\} dom g \subseteq A$
8		eqid	$⊢ \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\} = \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\}$
9	8	wfrlem3OLD	$⊢ g \in \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\} \to dom g \subseteq A$
10	7 9	mprgbir	$⊢ ⋃_{g \in \{f \| \exists x (f Fn x \land (x \subseteq A \land \forall y \in x Pred (R, A, y) \subseteq x) \land \forall y \in x f (y) = G ({f ↾}_{Pred (R, A, y)}))\}} dom g \subseteq A$
11	6 10	eqsstri	$⊢ dom F \subseteq A$