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Theorem resco 5085
 Description: Associative law for the restriction of a composition. (Contributed by set.mm contributors, 12-Dec-2006.)
Assertion
Ref Expression
resco ((A B) C) = (A (B C))

Proof of Theorem resco
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brco 4883 . . . . 5 (x(A B)yz(xBz zAy))
21anbi1i 676 . . . 4 ((x(A B)y x C) ↔ (z(xBz zAy) x C))
3 19.41v 1901 . . . 4 (z((xBz zAy) x C) ↔ (z(xBz zAy) x C))
4 an32 773 . . . . . 6 (((xBz zAy) x C) ↔ ((xBz x C) zAy))
5 brres 4949 . . . . . . 7 (x(B C)z ↔ (xBz x C))
65anbi1i 676 . . . . . 6 ((x(B C)z zAy) ↔ ((xBz x C) zAy))
74, 6bitr4i 243 . . . . 5 (((xBz zAy) x C) ↔ (x(B C)z zAy))
87exbii 1582 . . . 4 (z((xBz zAy) x C) ↔ z(x(B C)z zAy))
92, 3, 83bitr2i 264 . . 3 ((x(A B)y x C) ↔ z(x(B C)z zAy))
10 brres 4949 . . 3 (x((A B) C)y ↔ (x(A B)y x C))
11 brco 4883 . . 3 (x(A (B C))yz(x(B C)z zAy))
129, 10, 113bitr4i 268 . 2 (x((A B) C)yx(A (B C))y)
1312eqbrriv 4851 1 ((A B) C) = (A (B C))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   class class class wbr 4639   ∘ ccom 4721   ↾ 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-co 4726  df-xp 4784  df-res 4788 This theorem is referenced by:  coires1  5096  fcoi1  5240
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