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Metamath Proof Explorer

Theorem xpsbas

Description: The base set of the binary structure product. (Contributed by Mario Carneiro, 15-Aug-2015)

		Ref	Expression
	Hypotheses	xpsval.t	$⊢ T = R \times_{𝑠} S$
		xpsval.x	$⊢ X = {Base}_{R}$
		xpsval.y	$⊢ Y = {Base}_{S}$
		xpsval.1	$⊢ φ \to R \in V$
		xpsval.2	$⊢ φ \to S \in W$
	Assertion	xpsbas	$⊢ φ \to X \times Y = {Base}_{T}$

Proof

Step	Hyp	Ref	Expression
1		xpsval.t	$⊢ T = R \times_{𝑠} S$
2		xpsval.x	$⊢ X = {Base}_{R}$
3		xpsval.y	$⊢ Y = {Base}_{S}$
4		xpsval.1	$⊢ φ \to R \in V$
5		xpsval.2	$⊢ φ \to S \in W$
6		eqid	$⊢ (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) = (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})$
7		eqid	$⊢ Scalar (R) = Scalar (R)$
8		eqid	$⊢ Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\} = Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\}$
9	1 2 3 4 5 6 7 8	xpsval	$⊢ φ \to T = {(x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} “_{𝑠} (Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\})$
10	1 2 3 4 5 6 7 8	xpsrnbas	$⊢ φ \to ran (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) = {Base}_{(Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\})}$
11	6	xpsff1o2	$⊢ (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) : X \times Y \underset{1-1 onto}{⟶} ran (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})$
12		f1ocnv	$⊢ (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) : X \times Y \underset{1-1 onto}{⟶} ran (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \to {(x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} : ran (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \underset{1-1 onto}{⟶} X \times Y$
13	11 12	ax-mp	$⊢ {(x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} : ran (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \underset{1-1 onto}{⟶} X \times Y$
14		f1ofo	$⊢ {(x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} : ran (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \underset{1-1 onto}{⟶} X \times Y \to {(x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} : ran (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \underset{onto}{⟶} X \times Y$
15	13 14	mp1i	$⊢ φ \to {(x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\})}^{-1} : ran (x \in X, y \in Y ⟼ \{⟨\emptyset, x⟩, ⟨1_{𝑜}, y⟩\}) \underset{onto}{⟶} X \times Y$
16		ovexd	$⊢ φ \to Scalar (R) ⨉_{𝑠} \{⟨\emptyset, R⟩, ⟨1_{𝑜}, S⟩\} \in V$
17	9 10 15 16	imasbas	$⊢ φ \to X \times Y = {Base}_{T}$