Metamath Proof Explorer

Theorem csbima12gALTVD

Description: Virtual deduction proof of csbima12 . 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. csbima12 is csbima12gALTVD without virtual deductions and was automatically derived from csbima12gALTVD .

1:: |- (. A e. C ->. A e. C ).
2:1: ` |- (. A e. C ->. [_ A / x ]_ ( F |`B ) = ( ` [_ A / x ]_ F |`[_ A / x ]_ B ) ).
3:2: |- (. A e. C ->. ` ran [_ A / x ]_ ( F |`B ) ` = ran ( [_ A / x ]_ F |`[_ A / x ]_ B ) ).
4:1: |- (. A e. C ->. ` [_ A / x ]_ ran ( F |`B ) ` = ran [_ A / x ]_ ( F |`B ) ).
5:3,4: |- (. A e. C ->. ` [_ A / x ]_ ran ( F |`B ) ` = ran ( [_ A / x ]_ F |`[_ A / x ]_ B ) ).
6:: ` |- ( F " B ) = ran ( F |`B )
7:6: ` |- A. x ( F " B ) = ran ( F |`B )
8:1,7: |- (. A e. C ->. [_ A / x ]_ ( F " B ) = [_ ` A / x ]_ ran ( F |`B ) ).
9:5,8: |- (. A e. C ->. [_ A / x ]_ ( F " B ) = ` ran ( [_ A / x ]_ F |`[_ A / x ]_ B ) ).
10:: |- ( [_ A / x ]_ F " [_ A / x ]_ B ) = ran ` ( [_ A / x ]_ F |`[_ A / x ]_ B )
11:9,10: |- (. A e. C ->. [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) ).
qed:11: |- ( A e. C -> [_ A / x ]_ ( F " B ) = ( [_ A / x ]_ F " [_ A / x ]_ B ) )
(Contributed by Alan Sare, 10-Nov-2012) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion csbima12gALTVD ( 𝐴 ∈ 𝐶 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 “ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 idn1 ⊢ (    𝐴 ∈ 𝐶    ▶    𝐴 ∈ 𝐶    )
2 csbres ⊢ ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
3 2 a1i ⊢ ( 𝐴 ∈ 𝐶 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) )
4 1 3 e1a ⊢ (    𝐴 ∈ 𝐶    ▶    ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )    )
5 rneq ⊢ ( ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) → ran ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) )
6 4 5 e1a ⊢ (    𝐴 ∈ 𝐶    ▶    ran ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )    )
7 csbrn ⊢ ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 )
8 7 a1i ⊢ ( 𝐴 ∈ 𝐶 → ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) )
9 1 8 e1a ⊢ (    𝐴 ∈ 𝐶    ▶    ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 )    )
10 eqeq2 ⊢ ( ran ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) → ( ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) )
11 10 biimpd ⊢ ( ran ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) → ( ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 ↾ 𝐵 ) → ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) )
12 6 9 11 e11 ⊢ (    𝐴 ∈ 𝐶    ▶    ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )    )
13 df-ima ⊢ ( 𝐹 “ 𝐵 ) = ran ( 𝐹 ↾ 𝐵 )
14 13 ax-gen ⊢ ∀ 𝑥 ( 𝐹 “ 𝐵 ) = ran ( 𝐹 ↾ 𝐵 )
15 csbeq2 ⊢ ( ∀ 𝑥 ( 𝐹 “ 𝐵 ) = ran ( 𝐹 ↾ 𝐵 ) → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) )
16 15 a1i ⊢ ( 𝐴 ∈ 𝐶 → ( ∀ 𝑥 ( 𝐹 “ 𝐵 ) = ran ( 𝐹 ↾ 𝐵 ) → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) ) )
17 1 14 16 e10 ⊢ (    𝐴 ∈ 𝐶    ▶    ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 )    )
18 eqeq2 ⊢ ( ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) → ( ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) )
19 18 biimpd ⊢ ( ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) → ( ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ⦋ 𝐴 / 𝑥 ⦌ ran ( 𝐹 ↾ 𝐵 ) → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) )
20 12 17 19 e11 ⊢ (    𝐴 ∈ 𝐶    ▶    ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )    )
21 df-ima ⊢ ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 “ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )
22 eqeq2 ⊢ ( ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 “ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) → ( ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 “ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ↔ ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) )
23 22 biimprcd ⊢ ( ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) → ( ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 “ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) = ran ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 ↾ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 “ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) ) )
24 20 21 23 e10 ⊢ (    𝐴 ∈ 𝐶    ▶    ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 “ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 )    )
25 24 in1 ⊢ ( 𝐴 ∈ 𝐶 → ⦋ 𝐴 / 𝑥 ⦌ ( 𝐹 “ 𝐵 ) = ( ⦋ 𝐴 / 𝑥 ⦌ 𝐹 “ ⦋ 𝐴 / 𝑥 ⦌ 𝐵 ) )