Theorem breng

Description: Equinumerosity relation. This variation of bren does not require the Axiom of Union. (Contributed by BTernaryTau, 23-Sep-2024)

	Ref	Expression
Assertion	breng	$$⊢\left(A\in V\wedge B\in W\right)\to \left(A\approx B\leftrightarrow \exists ff:A\underset{1-1\; onto}{⟶}B\right)$$

Proof

Step	Hyp	Ref	Expression
1		f1oeq2	$$⊢x=A\to \left(f:x\underset{1-1\; onto}{⟶}y\leftrightarrow f:A\underset{1-1\; onto}{⟶}y\right)$$
2	1	exbidv	$$⊢x=A\to \left(\exists ff:x\underset{1-1\; onto}{⟶}y\leftrightarrow \exists ff:A\underset{1-1\; onto}{⟶}y\right)$$
3		f1oeq3	$$⊢y=B\to \left(f:A\underset{1-1\; onto}{⟶}y\leftrightarrow f:A\underset{1-1\; onto}{⟶}B\right)$$
4	3	exbidv	$$⊢y=B\to \left(\exists ff:A\underset{1-1\; onto}{⟶}y\leftrightarrow \exists ff:A\underset{1-1\; onto}{⟶}B\right)$$
5		df-en	$$⊢\approx =\left\{⟨x,y⟩|\exists ff:x\underset{1-1\; onto}{⟶}y\right\}$$
6	2 4 5	brabg	$$⊢\left(A\in V\wedge B\in W\right)\to \left(A\approx B\leftrightarrow \exists ff:A\underset{1-1\; onto}{⟶}B\right)$$