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Theorem cores 5084
 Description: Restricted first member of a class composition. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 12-Oct-2004.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
cores (ran B C → ((A C) B) = (A B))

Proof of Theorem cores
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brelrn 4960 . . . . . 6 (zByy ran B)
2 ssel 3267 . . . . . 6 (ran B C → (y ran By C))
3 iba 489 . . . . . . 7 (y C → (yAx ↔ (yAx y C)))
4 brres 4949 . . . . . . 7 (y(A C)x ↔ (yAx y C))
53, 4syl6rbbr 255 . . . . . 6 (y C → (y(A C)xyAx))
61, 2, 5syl56 30 . . . . 5 (ran B C → (zBy → (y(A C)xyAx)))
76pm5.32d 620 . . . 4 (ran B C → ((zBy y(A C)x) ↔ (zBy yAx)))
87exbidv 1626 . . 3 (ran B C → (y(zBy y(A C)x) ↔ y(zBy yAx)))
98opabbidv 4625 . 2 (ran B C → {z, x y(zBy y(A C)x)} = {z, x y(zBy yAx)})
10 df-co 4726 . 2 ((A C) B) = {z, x y(zBy y(A C)x)}
11 df-co 4726 . 2 (A B) = {z, x y(zBy yAx)}
129, 10, 113eqtr4g 2410 1 (ran B C → ((A C) B) = (A B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  {copab 4622   class class class wbr 4639   ∘ ccom 4721  ran crn 4773   ↾ 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-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788 This theorem is referenced by:  cores2  5091  fcoi2  5241
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