Theorem csbrngVD 42484

Description: Virtual deduction proof of csbrn 6104. 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 6104 is csbrngVD 42484 without virtual deductions and was automatically derived from csbrngVD 42484.

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 42162	. . . . . . . . . . . 12 ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2		sbcel12 4348	. . . . . . . . . . . . 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 42215	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
5		csbconstg 3856	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
6	1, 5	e1a 42215	. . . . . . . . . . . 12 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩   )
7		eleq1 2828	. . . . . . . . . . . 12 ⊢ (⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩ → (⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
8	6, 7	e1a 42215	. . . . . . . . . . 11 ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
9		bibi1 352	. . . . . . . . . . . 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 247	. . . . . . . . . . 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 42276	. . . . . . . . . 10 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
12	11	gen11 42204	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑤([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
13		exbi 1853	. . . . . . . . 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 42215	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
15		sbcex2 3786	. . . . . . . . . . 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 222	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → (∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵))
18	1, 17	e1a 42215	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   (∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵)   )
19		bitr3 353	. . . . . . . . 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 42276	. . . . . . 7 ⊢ (   𝐴 ∈ 𝑉   ▶   ([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
22	21	gen11 42204	. . . . . 6 ⊢ (   𝐴 ∈ 𝑉   ▶   ∀𝑦([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵)   )
23		abbi 2812	. . . . . . 7 ⊢ (∀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})
24	23	biimpi 215	. . . . . 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 42215	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
26		csbab 4377	. . . . . . 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 42215	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   )
29		eqeq2 2752	. . . . . 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 228	. . . . 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 42276	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
32		dfrn3 5796	. . . . . 6 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}
33	32	ax-gen 1802	. . . . 5 ⊢ ∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}
34		csbeq2 3842	. . . . . 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 42282	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}   )
37		eqeq2 2752	. . . . 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 228	. . . 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 42276	. . 3 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}   )
40		dfrn3 5796	. . 3 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
41		eqeq2 2752	. . . 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 249	. . 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 42282	. 2 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵   )
44	43	in1 42159	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1540   = wceq 1542  ∃wex 1786   ∈ wcel 2110  {cab 2717  [wsbc 3720  ⦋csb 3837  ⟨cop 4573  ran crn 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-cnv 5597  df-dm 5599  df-rn 5600  df-vd1 42158

This theorem is referenced by: (None)