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Theorem csbrngVD 39881
Description: Virtual deduction proof of csbrn 5811. 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 5811 is csbrngVD 39881 without virtual deductions and was automatically derived from csbrngVD 39881.
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 39549 . . . . . . . . . . . 12 (   𝐴𝑉   ▶   𝐴𝑉   )
2 sbcel12 4177 . . . . . . . . . . . . 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 39611 . . . . . . . . . . 11 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
5 csbconstg 3740 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 / 𝑥𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
61, 5e1a 39611 . . . . . . . . . . . 12 (   𝐴𝑉   ▶   𝐴 / 𝑥𝑤, 𝑦⟩ = ⟨𝑤, 𝑦   )
7 eleq1 2865 . . . . . . . . . . . 12 (𝐴 / 𝑥𝑤, 𝑦⟩ = ⟨𝑤, 𝑦⟩ → (𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
86, 7e1a 39611 . . . . . . . . . . 11 (   𝐴𝑉   ▶   (𝐴 / 𝑥𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
9 bibi1 343 . . . . . . . . . . . 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 240 . . . . . . . . . . 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 39672 . . . . . . . . . 10 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
1211gen11 39600 . . . . . . . . 9 (   𝐴𝑉   ▶   𝑤([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
13 exbi 1943 . . . . . . . . 9 (∀𝑤([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) → (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
1412, 13e1a 39611 . . . . . . . 8 (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
15 sbcex2 3683 . . . . . . . . . . 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 215 . . . . . . . . 9 (𝐴𝑉 → (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵))
181, 17e1a 39611 . . . . . . . 8 (   𝐴𝑉   ▶   (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵)   )
19 bitr3 344 . . . . . . . . 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 39672 . . . . . . 7 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
2221gen11 39600 . . . . . 6 (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵)   )
23 abbi 2913 . . . . . . 7 (∀𝑦([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) ↔ {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵})
2423biimpi 208 . . . . . 6 (∀𝑦([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵) → {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵})
2522, 24e1a 39611 . . . . 5 (   𝐴𝑉   ▶   {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
26 csbab 4203 . . . . . . 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 39611 . . . . 5 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵}   )
29 eqeq2 2809 . . . . . 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 221 . . . . 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 39672 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
32 dfrn3 5514 . . . . . 6 ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}
3332ax-gen 1891 . . . . 5 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}
34 csbeq2 3731 . . . . . 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 39678 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}   )
37 eqeq2 2809 . . . . 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 221 . . . 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 39672 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}   )
40 dfrn3 5514 . . 3 ran 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}
41 eqeq2 2809 . . . 4 (ran 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵} → (𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵𝐴 / 𝑥ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}))
4241biimprcd 242 . . 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 39678 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵   )
4443in1 39546 1 (𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651   = wceq 1653  wex 1875  wcel 2157  {cab 2784  [wsbc 3632  csb 3727  cop 4373  ran crn 5312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-br 4843  df-opab 4905  df-cnv 5319  df-dm 5321  df-rn 5322  df-vd1 39545
This theorem is referenced by: (None)
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