Theorem rescnvcnv 3499

Description: The restriction of the double converse of a class.

Assertion

Ref	Expression
rescnvcnv	⊢ (◡◡A ↾ B) = (A ↾ B)

Proof of Theorem rescnvcnv

Step	Hyp	Ref	Expression
1		cnvcnv 3492	. . . 4 ⊢ ◡◡A = (A ∩ (V × V))
2	1	ineq1i 2216	. . 3 ⊢ (◡◡A ∩ (B × V)) = ((A ∩ (V × V)) ∩ (B × V))
3		inass 2226	. . 3 ⊢ ((A ∩ (V × V)) ∩ (B × V)) = (A ∩ ((V × V) ∩ (B × V)))
4		inxp 3275	. . . . 5 ⊢ ((V × V) ∩ (B × V)) = ((V ∩ B) × (V ∩ V))
5		incom 2211	. . . . . . 7 ⊢ (V ∩ B) = (B ∩ V)
6		inv1 2303	. . . . . . 7 ⊢ (B ∩ V) = B
7	5, 6	eqtr 1498	. . . . . 6 ⊢ (V ∩ B) = B
8		xpeq1 3206	. . . . . 6 ⊢ ((V ∩ B) = B → ((V ∩ B) × (V ∩ V)) = (B × (V ∩ V)))
9	7, 8	ax-mp 7	. . . . 5 ⊢ ((V ∩ B) × (V ∩ V)) = (B × (V ∩ V))
10		inidm 2225	. . . . . 6 ⊢ (V ∩ V) = V
11		xpeq2 3207	. . . . . 6 ⊢ ((V ∩ V) = V → (B × (V ∩ V)) = (B × V))
12	10, 11	ax-mp 7	. . . . 5 ⊢ (B × (V ∩ V)) = (B × V)
13	4, 9, 12	3eqtr 1502	. . . 4 ⊢ ((V × V) ∩ (B × V)) = (B × V)
14	13	ineq2i 2217	. . 3 ⊢ (A ∩ ((V × V) ∩ (B × V))) = (A ∩ (B × V))
15	2, 3, 14	3eqtr 1502	. 2 ⊢ (◡◡A ∩ (B × V)) = (A ∩ (B × V))
16		df-res 3196	. 2 ⊢ (◡◡A ↾ B) = (◡◡A ∩ (B × V))
17		df-res 3196	. 2 ⊢ (A ↾ B) = (A ∩ (B × V))
18	15, 16, 17	3eqtr4 1508	1 ⊢ (◡◡A ↾ B) = (A ↾ B)

Syntax hints:   = wceq 958  Vcvv 1814   ∩ cin 2049   × cxp 3174  ^◡ccnv 3175   ↾ cres 3178

This theorem is referenced by:  cnvcnvres 3500  imacnvcnv 3501  resdm2 3502  resdmres 3503

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-res 3196

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