Metamath Proof Explorer

Theorem csbrngVD

Description: Virtual deduction proof of csbrn . The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbrn is csbrngVD without virtual deductions and was automatically derived from csbrngVD .

1:: |- (. A e. V ->. A e. V ).
2:1: |- (. A e. V ->. ( [. A / x ]. <. w ,. y >. e. B <-> [_ A / x ]_ <. w , y >. e. [_ A / x ]_ B ) ).
3:1: |- (. A e. V ->. [_ A / x ]_ <. w ,. y >. = <. w , y >. ).
4:3: |- (. A e. V ->. ( [_ A / x ]_ <. w ,. y >. e. [_ A / x ]_ B <-> <. w , y >. e. [_ A / x ]_ B ) ).
5:2,4: |- (. A e. V ->. ( [. A / x ]. <. w ,. y >. e. B <-> <. w , y >. e. [_ A / x ]_ B ) ).
6:5: |- (. A e. V ->. A. w ( [. A / x ]. <. w ,. y >. e. B <-> <. w , y >. e. [_ A / x ]_ B ) ).
7:6: |- (. A e. V ->. ( E. w [. A / x ]. <. w ,. y >. e. B <-> E. w <. w , y >. e. [_ A / x ]_ B ) ).
8:1: |- (. A e. V ->. ( E. w [. A / x ]. <. w ,. y >. e. B <-> [. A / x ]. E. w <. w , y >. e. B ) ).
9:7,8: |- (. A e. V ->. ( [. A / x ]. E. w <. w ,. y >. e. B <-> E. w <. w , y >. e. [_ A / x ]_ B ) ).
10:9: |- (. A e. V ->. A. y ( [. A / x ]. E. w <. w , y >. e. B <-> E. w <. w , y >. e. [_ A / x ]_ B ) ).
11:10: |- (. A e. V ->. { y | [. A / x ]. E. w <. w , y >. e. B } = { y | E. w <. w , y >. e. [_ A / x ]_ B } ).
12:1: |- (. A e. V ->. [_ A / x ]_ { y | E. w <. w , y >. e. B } = { y | [. A / x ]. E. w <. w , y >. e. B } ).
13:11,12: |- (. A e. V ->. [_ A / x ]_ { y | E. w <. w , y >. e. B } = { y | E. w <. w , y >. e. [_ A / x ]_ B } ).
14:: |- ran B = { y | E. w <. w ,. y >. e. B }
15:14: |- A. x ran B = { y | E. w <. w ,. y >. e. B }
16:1,15: |- (. A e. V ->. [_ A / x ]_ ran B = [_ A / x ]_ { y | E. w <. w , y >. e. B } ).
17:13,16: |- (. A e. V ->. [_ A / x ]_ ran B = { y | E. w <. w , y >. e. [_ A / x ]_ B } ).
18:: |- ran [_ A / x ]_ B = { y | E. w <. w ,. y >. e. [_ A / x ]_ B }
19:17,18: |- (. A e. V ->. [_ A / x ]_ ran B = ran [_ A / x ]_ B ).
qed:19: |- ( A e. V -> [_ A / x ]_ ran B = ran [_ A / x ]_ B )
(Contributed by Alan Sare, 10-Nov-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion csbrngVD ⊢A∈V→⦋A/x⦌ran⁡B=ran⁡⦋A/x⦌B

Proof

Step Hyp Ref Expression
1 idn1 ⊢A∈V→A∈V
2 sbcel12 ⊢[˙A/x]˙wy∈B↔⦋A/x⦌wy∈⦋A/x⦌B
3 2 a1i ⊢A∈V→[˙A/x]˙wy∈B↔⦋A/x⦌wy∈⦋A/x⦌B
4 1 3 e1a ⊢A∈V→[˙A/x]˙wy∈B↔⦋A/x⦌wy∈⦋A/x⦌B
5 csbconstg ⊢A∈V→⦋A/x⦌wy=wy
6 1 5 e1a ⊢A∈V→⦋A/x⦌wy=wy
7 eleq1 ⊢⦋A/x⦌wy=wy→⦋A/x⦌wy∈⦋A/x⦌B↔wy∈⦋A/x⦌B
8 6 7 e1a ⊢A∈V→⦋A/x⦌wy∈⦋A/x⦌B↔wy∈⦋A/x⦌B
9 bibi1 ⊢[˙A/x]˙wy∈B↔⦋A/x⦌wy∈⦋A/x⦌B→[˙A/x]˙wy∈B↔wy∈⦋A/x⦌B↔⦋A/x⦌wy∈⦋A/x⦌B↔wy∈⦋A/x⦌B
10 9 biimprd ⊢[˙A/x]˙wy∈B↔⦋A/x⦌wy∈⦋A/x⦌B→⦋A/x⦌wy∈⦋A/x⦌B↔wy∈⦋A/x⦌B→[˙A/x]˙wy∈B↔wy∈⦋A/x⦌B
11 4 8 10 e11 ⊢A∈V→[˙A/x]˙wy∈B↔wy∈⦋A/x⦌B
12 11 gen11 ⊢A∈V→∀w[˙A/x]˙wy∈B↔wy∈⦋A/x⦌B
13 exbi ⊢∀w[˙A/x]˙wy∈B↔wy∈⦋A/x⦌B→∃w[˙A/x]˙wy∈B↔∃wwy∈⦋A/x⦌B
14 12 13 e1a ⊢A∈V→∃w[˙A/x]˙wy∈B↔∃wwy∈⦋A/x⦌B
15 sbcex2 ⊢[˙A/x]˙∃wwy∈B↔∃w[˙A/x]˙wy∈B
16 15 a1i ⊢A∈V→[˙A/x]˙∃wwy∈B↔∃w[˙A/x]˙wy∈B
17 16 bicomd ⊢A∈V→∃w[˙A/x]˙wy∈B↔[˙A/x]˙∃wwy∈B
18 1 17 e1a ⊢A∈V→∃w[˙A/x]˙wy∈B↔[˙A/x]˙∃wwy∈B
19 bitr3 ⊢∃w[˙A/x]˙wy∈B↔[˙A/x]˙∃wwy∈B→∃w[˙A/x]˙wy∈B↔∃wwy∈⦋A/x⦌B→[˙A/x]˙∃wwy∈B↔∃wwy∈⦋A/x⦌B
20 19 com12 ⊢∃w[˙A/x]˙wy∈B↔∃wwy∈⦋A/x⦌B→∃w[˙A/x]˙wy∈B↔[˙A/x]˙∃wwy∈B→[˙A/x]˙∃wwy∈B↔∃wwy∈⦋A/x⦌B
21 14 18 20 e11 ⊢A∈V→[˙A/x]˙∃wwy∈B↔∃wwy∈⦋A/x⦌B
22 21 gen11 ⊢A∈V→∀y[˙A/x]˙∃wwy∈B↔∃wwy∈⦋A/x⦌B
23 abbib ⊢y|[˙A/x]˙∃wwy∈B=y|∃wwy∈⦋A/x⦌B↔∀y[˙A/x]˙∃wwy∈B↔∃wwy∈⦋A/x⦌B
24 23 biimpri ⊢∀y[˙A/x]˙∃wwy∈B↔∃wwy∈⦋A/x⦌B→y|[˙A/x]˙∃wwy∈B=y|∃wwy∈⦋A/x⦌B
25 22 24 e1a ⊢A∈V→y|[˙A/x]˙∃wwy∈B=y|∃wwy∈⦋A/x⦌B
26 csbab ⊢⦋A/x⦌y|∃wwy∈B=y|[˙A/x]˙∃wwy∈B
27 26 a1i ⊢A∈V→⦋A/x⦌y|∃wwy∈B=y|[˙A/x]˙∃wwy∈B
28 1 27 e1a ⊢A∈V→⦋A/x⦌y|∃wwy∈B=y|[˙A/x]˙∃wwy∈B
29 eqeq2 ⊢y|[˙A/x]˙∃wwy∈B=y|∃wwy∈⦋A/x⦌B→⦋A/x⦌y|∃wwy∈B=y|[˙A/x]˙∃wwy∈B↔⦋A/x⦌y|∃wwy∈B=y|∃wwy∈⦋A/x⦌B
30 29 biimpd ⊢y|[˙A/x]˙∃wwy∈B=y|∃wwy∈⦋A/x⦌B→⦋A/x⦌y|∃wwy∈B=y|[˙A/x]˙∃wwy∈B→⦋A/x⦌y|∃wwy∈B=y|∃wwy∈⦋A/x⦌B
31 25 28 30 e11 ⊢A∈V→⦋A/x⦌y|∃wwy∈B=y|∃wwy∈⦋A/x⦌B
32 dfrn3 ⊢ran⁡B=y|∃wwy∈B
33 32 ax-gen ⊢∀xran⁡B=y|∃wwy∈B
34 csbeq2 ⊢∀xran⁡B=y|∃wwy∈B→⦋A/x⦌ran⁡B=⦋A/x⦌y|∃wwy∈B
35 34 a1i ⊢A∈V→∀xran⁡B=y|∃wwy∈B→⦋A/x⦌ran⁡B=⦋A/x⦌y|∃wwy∈B
36 1 33 35 e10 ⊢A∈V→⦋A/x⦌ran⁡B=⦋A/x⦌y|∃wwy∈B
37 eqeq2 ⊢⦋A/x⦌y|∃wwy∈B=y|∃wwy∈⦋A/x⦌B→⦋A/x⦌ran⁡B=⦋A/x⦌y|∃wwy∈B↔⦋A/x⦌ran⁡B=y|∃wwy∈⦋A/x⦌B
38 37 biimpd ⊢⦋A/x⦌y|∃wwy∈B=y|∃wwy∈⦋A/x⦌B→⦋A/x⦌ran⁡B=⦋A/x⦌y|∃wwy∈B→⦋A/x⦌ran⁡B=y|∃wwy∈⦋A/x⦌B
39 31 36 38 e11 ⊢A∈V→⦋A/x⦌ran⁡B=y|∃wwy∈⦋A/x⦌B
40 dfrn3 ⊢ran⁡⦋A/x⦌B=y|∃wwy∈⦋A/x⦌B
41 eqeq2 ⊢ran⁡⦋A/x⦌B=y|∃wwy∈⦋A/x⦌B→⦋A/x⦌ran⁡B=ran⁡⦋A/x⦌B↔⦋A/x⦌ran⁡B=y|∃wwy∈⦋A/x⦌B
42 41 biimprcd ⊢⦋A/x⦌ran⁡B=y|∃wwy∈⦋A/x⦌B→ran⁡⦋A/x⦌B=y|∃wwy∈⦋A/x⦌B→⦋A/x⦌ran⁡B=ran⁡⦋A/x⦌B
43 39 40 42 e10 ⊢A∈V→⦋A/x⦌ran⁡B=ran⁡⦋A/x⦌B
44 43 in1 ⊢A∈V→⦋A/x⦌ran⁡B=ran⁡⦋A/x⦌B