Theorem csbresg 4977

Description: Distribute proper substitution through the restriction of a class. csbresg 4977 is derived from the virtual deduction proof csbresgVD in set.mm. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

Ref	Expression
csbresg	⊢ (A ∈ V → [A / x](B ↾ C) = ([A / x]B ↾ [A / x]C))

Proof of Theorem csbresg

Step	Hyp	Ref	Expression
1		csbing 3463	. . 3 ⊢ (A ∈ V → [A / x](B ∩ (C × V)) = ([A / x]B ∩ [A / x](C × V)))
2		csbxpg 4814	. . . . 5 ⊢ (A ∈ V → [A / x](C × V) = ([A / x]C × [A / x]V))
3		csbconstg 3151	. . . . . 6 ⊢ (A ∈ V → [A / x]V = V)
4	3	xpeq2d 4809	. . . . 5 ⊢ (A ∈ V → ([A / x]C × [A / x]V) = ([A / x]C × V))
5	2, 4	eqtrd 2385	. . . 4 ⊢ (A ∈ V → [A / x](C × V) = ([A / x]C × V))
6	5	ineq2d 3458	. . 3 ⊢ (A ∈ V → ([A / x]B ∩ [A / x](C × V)) = ([A / x]B ∩ ([A / x]C × V)))
7	1, 6	eqtrd 2385	. 2 ⊢ (A ∈ V → [A / x](B ∩ (C × V)) = ([A / x]B ∩ ([A / x]C × V)))
8		df-res 4789	. . 3 ⊢ (B ↾ C) = (B ∩ (C × V))
9	8	csbeq2i 3163	. 2 ⊢ [A / x](B ↾ C) = [A / x](B ∩ (C × V))
10		df-res 4789	. 2 ⊢ ([A / x]B ↾ [A / x]C) = ([A / x]B ∩ ([A / x]C × V))
11	7, 9, 10	3eqtr4g 2410	1 ⊢ (A ∈ V → [A / x](B ↾ C) = ([A / x]B ↾ [A / x]C))

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2860  [csb 3137   ∩ cin 3209   × cxp 4771   ↾ cres 4775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-opab 4624  df-xp 4785  df-res 4789

This theorem is referenced by: (None)