Theorem csbresgVD 43641

Description: Virtual deduction proof of csbres 5982. 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. csbres 5982 is csbresgVD 43641 without virtual deductions and was automatically derived from csbresgVD 43641.

1::	⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌V = V   )
3:2:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   )
4:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)   )
5:3,4:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   )
6:5:	⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   )
7:1:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))   )
8:6,7:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   )
9::	⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
10:9:	⊢ ∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
11:1,10:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V))   )
12:8,11:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   )
13::	⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))
14:12,13:	⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)   )
qed:14:	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ( ⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶))

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

Assertion

Ref	Expression
csbresgVD	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶))

Proof of Theorem csbresgVD

Step	Hyp	Ref	Expression
1		idn1 43320	. . . . . . . . 9 ⊢ (   𝐴 ∈ 𝑉   ▶   𝐴 ∈ 𝑉   )
2		csbconstg 3911	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌V = V)
3	1, 2	e1a 43373	. . . . . . . 8 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌V = V   )
4		xpeq2 5696	. . . . . . . 8 ⊢ (⦋𝐴 / 𝑥⦌V = V → (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V))
5	3, 4	e1a 43373	. . . . . . 7 ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   )
6		csbxp 5773	. . . . . . . . 9 ⊢ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)
7	6	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V))
8	1, 7	e1a 43373	. . . . . . 7 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V)   )
9		eqeq2 2744	. . . . . . . 8 ⊢ ((⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V) → (⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) ↔ ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)))
10	9	biimpd 228	. . . . . . 7 ⊢ ((⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐶 × V) → (⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × ⦋𝐴 / 𝑥⦌V) → ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)))
11	5, 8, 10	e11 43434	. . . . . 6 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V)   )
12		ineq2 4205	. . . . . 6 ⊢ (⦋𝐴 / 𝑥⦌(𝐶 × V) = (⦋𝐴 / 𝑥⦌𝐶 × V) → (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)))
13	11, 12	e1a 43373	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   )
14		csbin 4438	. . . . . . 7 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))
15	14	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)))
16	1, 15	e1a 43373	. . . . 5 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V))   )
17		eqeq2 2744	. . . . . 6 ⊢ ((⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) ↔ ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))))
18	17	biimpd 228	. . . . 5 ⊢ ((⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ ⦋𝐴 / 𝑥⦌(𝐶 × V)) → ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))))
19	13, 16, 18	e11 43434	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   )
20		df-res 5687	. . . . . 6 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
21	20	ax-gen 1797	. . . . 5 ⊢ ∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V))
22		csbeq2 3897	. . . . . 6 ⊢ (∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)))
23	22	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥(𝐵 ↾ 𝐶) = (𝐵 ∩ (𝐶 × V)) → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V))))
24	1, 21, 23	e10 43440	. . . 4 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V))   )
25		eqeq2 2744	. . . . 5 ⊢ (⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) ↔ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))))
26	25	biimpd 228	. . . 4 ⊢ (⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = ⦋𝐴 / 𝑥⦌(𝐵 ∩ (𝐶 × V)) → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))))
27	19, 24, 26	e11 43434	. . 3 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))   )
28		df-res 5687	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))
29		eqeq2 2744	. . . 4 ⊢ ((⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → (⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) ↔ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V))))
30	29	biimprcd 249	. . 3 ⊢ (⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → ((⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ∩ (⦋𝐴 / 𝑥⦌𝐶 × V)) → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)))
31	27, 28, 30	e10 43440	. 2 ⊢ (   𝐴 ∈ 𝑉   ▶   ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶)   )
32	31	in1 43317	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶))

Syntax hints:   → wi 4  ∀wal 1539   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ⦋csb 3892   ∩ cin 3946   × cxp 5673   ↾ cres 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-in 3954  df-nul 4322  df-opab 5210  df-xp 5681  df-res 5687  df-vd1 43316

This theorem is referenced by: (None)