Theorem resdmres 5079

Description: Restriction to the domain of a restriction. (Contributed by set.mm contributors, 8-Apr-2007.)

Assertion

Ref	Expression
resdmres	⊢ (A ↾ dom (A ↾ B)) = (A ↾ B)

Proof of Theorem resdmres

Step	Hyp	Ref	Expression
1		in12 3467	. . 3 ⊢ (A ∩ ((B × V) ∩ (dom A × V))) = ((B × V) ∩ (A ∩ (dom A × V)))
2		df-res 4789	. . . . 5 ⊢ (A ↾ dom A) = (A ∩ (dom A × V))
3		resdm 4999	. . . . 5 ⊢ (A ↾ dom A) = A
4	2, 3	eqtr3i 2375	. . . 4 ⊢ (A ∩ (dom A × V)) = A
5	4	ineq2i 3455	. . 3 ⊢ ((B × V) ∩ (A ∩ (dom A × V))) = ((B × V) ∩ A)
6		incom 3449	. . 3 ⊢ ((B × V) ∩ A) = (A ∩ (B × V))
7	1, 5, 6	3eqtri 2377	. 2 ⊢ (A ∩ ((B × V) ∩ (dom A × V))) = (A ∩ (B × V))
8		df-res 4789	. . 3 ⊢ (A ↾ dom (A ↾ B)) = (A ∩ (dom (A ↾ B) × V))
9		dmres 4987	. . . . . 6 ⊢ dom (A ↾ B) = (B ∩ dom A)
10	9	xpeq1i 4805	. . . . 5 ⊢ (dom (A ↾ B) × V) = ((B ∩ dom A) × V)
11		xpindir 4866	. . . . 5 ⊢ ((B ∩ dom A) × V) = ((B × V) ∩ (dom A × V))
12	10, 11	eqtri 2373	. . . 4 ⊢ (dom (A ↾ B) × V) = ((B × V) ∩ (dom A × V))
13	12	ineq2i 3455	. . 3 ⊢ (A ∩ (dom (A ↾ B) × V)) = (A ∩ ((B × V) ∩ (dom A × V)))
14	8, 13	eqtri 2373	. 2 ⊢ (A ↾ dom (A ↾ B)) = (A ∩ ((B × V) ∩ (dom A × V)))
15		df-res 4789	. 2 ⊢ (A ↾ B) = (A ∩ (B × V))
16	7, 14, 15	3eqtr4i 2383	1 ⊢ (A ↾ dom (A ↾ B)) = (A ↾ B)

Syntax hints:   = wceq 1642  Vcvv 2860   ∩ cin 3209   × cxp 4771  dom cdm 4773   ↾ 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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789

This theorem is referenced by:  imadmres  5080