Theorem csbrngVD 44987

Description: Virtual deduction proof of csbrn 6150. 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 6150 is csbrngVD 44987 without virtual deductions and was automatically derived from csbrngVD 44987.

1::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤   ,   𝑦⟩ ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
3:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌⟨𝑤   ,   𝑦⟩ = ⟨𝑤, 𝑦⟩   )
4:3:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌⟨𝑤   ,   𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
5:2,4:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤   ,   𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
6:5:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑤([𝐴 / 𝑥]⟨𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
7:6:	⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
8:1:	⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵)   )
9:7,8:	⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤⟨𝑤    ,   𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
10:9:	⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]∃𝑤 ⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
11:10:	⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨ 𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
12:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤 ⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   )
13:11,12:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤 ⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
14::	⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤   ,   𝑦⟩ ∈ 𝐵}
15:14:	⊢ ∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤   ,   𝑦⟩ ∈ 𝐵}
16:1,15:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   )
17:13,16:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
18::	⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤    ,   𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
19:17,18:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋ 𝐴 / 𝑥⦌𝐵   )
qed:19:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)

(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
csbrngVD	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)

Proof of Theorem csbrngVD

Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn1 44666	. . . . . . . . . . . 12 ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2		sbcel12 4358	. . . . . . . . . . . . 13 ⊢ ([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)
3	2	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
4	1, 3	e1a 44719	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
5		csbconstg 3864	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
6	1, 5	e1a 44719	. . . . . . . . . . . 12 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩   )
7		eleq1 2819	. . . . . . . . . . . 12 ⊢ (⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩ → (⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
8	6, 7	e1a 44719	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
9		bibi1 351	. . . . . . . . . . . 12 ⊢ (([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) → (([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) ↔ (⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)))
10	9	biimprd 248	. . . . . . . . . . 11 ⊢ (([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) → ((⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) → ([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)))
11	4, 8, 10	e11 44780	. . . . . . . . . 10 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
12	11	gen11 44708	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑤([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
13		exbi 1848	. . . . . . . . 9 ⊢ (∀𝑤([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) → (∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
14	12, 13	e1a 44719	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
15		sbcex2 3797	. . . . . . . . . . 11 ⊢ ([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵)
16	15	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵))
17	16	bicomd 223	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵))
18	1, 17	e1a 44719	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵)   )
19		bitr3 352	. . . . . . . . 9 ⊢ ((∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵) → ((∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) → ([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)))
20	19	com12 32	. . . . . . . 8 ⊢ ((∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) → ((∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵) → ([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)))
21	14, 18, 20	e11 44780	. . . . . . 7 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
22	21	gen11 44708	. . . . . 6 ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
23		abbib 2800	. . . . . . 7 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} ↔ ∀𝑦([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
24	23	biimpri 228	. . . . . 6 ⊢ (∀𝑦([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵) → {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵})
25	22, 24	e1a 44719	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
26		csbab 4387	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}
27	26	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵})
28	1, 27	e1a 44719	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   )
29		eqeq2 2743	. . . . . 6 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} ↔ ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}))
30	29	biimpd 229	. . . . 5 ⊢ ({𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}))
31	25, 28, 30	e11 44780	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
32		dfrn3 5828	. . . . . 6 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}
33	32	ax-gen 1796	. . . . 5 ⊢ ∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}
34		csbeq2 3850	. . . . . 6 ⊢ (∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} → ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵})
35	34	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} → ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}))
36	1, 33, 35	e10 44786	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   )
37		eqeq2 2743	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} ↔ ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}))
38	37	biimpd 229	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} → ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}))
39	31, 36, 38	e11 44780	. . 3 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
40		dfrn3 5828	. . 3 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
41		eqeq2 2743	. . . 4 ⊢ (ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} → (⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵 ↔ ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}))
42	41	biimprcd 250	. . 3 ⊢ (⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} → (ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵} → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵))
43	39, 40, 42	e10 44786	. 2 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵   )
44	43	in1 44663	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  [wsbc 3736  ⦋csb 3845  ⟨cop 4579  ran crn 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-cnv 5622  df-dm 5624  df-rn 5625  df-vd1 44662

This theorem is referenced by: (None)