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Theorem resdmres 5078
 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
StepHypRef Expression
1 in12 3466 . . 3 (A ∩ ((B × V) ∩ (dom A × V))) = ((B × V) ∩ (A ∩ (dom A × V)))
2 df-res 4788 . . . . 5 (A dom A) = (A ∩ (dom A × V))
3 resdm 4998 . . . . 5 (A dom A) = A
42, 3eqtr3i 2375 . . . 4 (A ∩ (dom A × V)) = A
54ineq2i 3454 . . 3 ((B × V) ∩ (A ∩ (dom A × V))) = ((B × V) ∩ A)
6 incom 3448 . . 3 ((B × V) ∩ A) = (A ∩ (B × V))
71, 5, 63eqtri 2377 . 2 (A ∩ ((B × V) ∩ (dom A × V))) = (A ∩ (B × V))
8 df-res 4788 . . 3 (A dom (A B)) = (A ∩ (dom (A B) × V))
9 dmres 4986 . . . . . 6 dom (A B) = (B ∩ dom A)
109xpeq1i 4804 . . . . 5 (dom (A B) × V) = ((B ∩ dom A) × V)
11 xpindir 4865 . . . . 5 ((B ∩ dom A) × V) = ((B × V) ∩ (dom A × V))
1210, 11eqtri 2373 . . . 4 (dom (A B) × V) = ((B × V) ∩ (dom A × V))
1312ineq2i 3454 . . 3 (A ∩ (dom (A B) × V)) = (A ∩ ((B × V) ∩ (dom A × V)))
148, 13eqtri 2373 . 2 (A dom (A B)) = (A ∩ ((B × V) ∩ (dom A × V)))
15 df-res 4788 . 2 (A B) = (A ∩ (B × V))
167, 14, 153eqtr4i 2383 1 (A dom (A B)) = (A B)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  Vcvv 2859   ∩ cin 3208   × cxp 4770  dom cdm 4772   ↾ cres 4774 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788 This theorem is referenced by:  imadmres  5079
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