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Theorem csbrngVD 44894
Description: Virtual deduction proof of csbrn 6225. 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 6225 is csbrngVD 44894 without virtual deductions and was automatically derived from csbrngVD 44894.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤   ,   𝑦 𝐵𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
3:1: (   𝐴𝑉   ▶   𝐴 / 𝑥𝑤   ,   𝑦⟩ = 𝑤, 𝑦   )
4:3: (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑤   ,   𝑦 𝐴 / 𝑥𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
5:2,4: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤   ,   𝑦 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
6:5: (   𝐴𝑉   ▶   𝑤([𝐴 / 𝑥]𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
7:6: (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑤   ,    𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
8:1: (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑤   ,    𝑦⟩ ∈ 𝐵[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵)   )
9:7,8: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑤    ,   𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
10:9: (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥]𝑤 𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
11:10: (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑤 𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
12:1: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦 ∣ ∃𝑤 𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵}   )
13:11,12: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦 ∣ ∃𝑤 𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
14:: ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤   ,   𝑦⟩ ∈ 𝐵}
15:14: 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤   ,   𝑦 𝐵}
16:1,15: (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}   )
17:13,16: (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = {𝑦 𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
18:: ran 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑤    ,   𝑦⟩ ∈ 𝐴 / 𝑥𝐵}
19:17,18: (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵   )
qed:19: (𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbrngVD (𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)

Proof of Theorem csbrngVD
Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 44572 . . . . . . . . . . . 12 (   𝐴𝑉   ▶   𝐴𝑉   )
2 sbcel12 4417 . . . . . . . . . . . . 13 ([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)
32a1i 11 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
41, 3e1a 44625 . . . . . . . . . . 11 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
5 csbconstg 3927 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 / 𝑥𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
61, 5e1a 44625 . . . . . . . . . . . 12 (   𝐴𝑉   ▶   𝐴 / 𝑥𝑤, 𝑦⟩ = ⟨𝑤, 𝑦   )
7 eleq1 2827 . . . . . . . . . . . 12 (𝐴 / 𝑥𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩ → (𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
86, 7e1a 44625 . . . . . . . . . . 11 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
9 bibi1 351 . . . . . . . . . . . 12 (([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) → (([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) ↔ (𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)))
109biimprd 248 . . . . . . . . . . 11 (([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) → ((𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) → ([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)))
114, 8, 10e11 44686 . . . . . . . . . 10 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
1211gen11 44614 . . . . . . . . 9 (   𝐴𝑉   ▶   𝑤([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
13 exbi 1844 . . . . . . . . 9 (∀𝑤([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) → (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
1412, 13e1a 44625 . . . . . . . 8 (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
15 sbcex2 3856 . . . . . . . . . . 11 ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵)
1615a1i 11 . . . . . . . . . 10 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵))
1716bicomd 223 . . . . . . . . 9 (𝐴𝑉 → (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵))
181, 17e1a 44625 . . . . . . . 8 (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵)   )
19 bitr3 352 . . . . . . . . 9 ((∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵) → ((∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) → ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)))
2019com12 32 . . . . . . . 8 ((∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) → ((∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵) → ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)))
2114, 18, 20e11 44686 . . . . . . 7 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
2221gen11 44614 . . . . . 6 (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
23 abbib 2809 . . . . . . 7 ({𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵} ↔ ∀𝑦([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
2423biimpri 228 . . . . . 6 (∀𝑦([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) → {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵})
2522, 24e1a 44625 . . . . 5 (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
26 csbab 4446 . . . . . . 7 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵}
2726a1i 11 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵})
281, 27e1a 44625 . . . . 5 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵}   )
29 eqeq2 2747 . . . . . 6 ({𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} ↔ 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}))
3029biimpd 229 . . . . 5 ({𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} → 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}))
3125, 28, 30e11 44686 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
32 dfrn3 5903 . . . . . 6 ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}
3332ax-gen 1792 . . . . 5 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}
34 csbeq2 3913 . . . . . 6 (∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} → 𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵})
3534a1i 11 . . . . 5 (𝐴𝑉 → (∀𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} → 𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}))
361, 33, 35e10 44692 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}   )
37 eqeq2 2747 . . . . 5 (𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} ↔ 𝐴 / 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}))
3837biimpd 229 . . . 4 (𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} → 𝐴 / 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}))
3931, 36, 38e11 44686 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
40 dfrn3 5903 . . 3 ran 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}
41 eqeq2 2747 . . . 4 (ran 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵𝐴 / 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}))
4241biimprcd 250 . . 3 (𝐴 / 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵} → (ran 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵} → 𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵))
4339, 40, 42e10 44692 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵   )
4443in1 44569 1 (𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  [wsbc 3791  csb 3908  cop 4637  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700  df-vd1 44568
This theorem is referenced by: (None)
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