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Theorem dmcosseq 4973
 Description: Domain of a composition. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 28-May-1998.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
dmcosseq (ran B dom A → dom (A B) = dom B)

Proof of Theorem dmcosseq
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmcoss 4971 . . 3 dom (A B) dom B
21a1i 10 . 2 (ran B dom A → dom (A B) dom B)
3 brelrn 4960 . . . . . . . . . 10 (xByy ran B)
4 ssel 3267 . . . . . . . . . 10 (ran B dom A → (y ran By dom A))
53, 4syl5 28 . . . . . . . . 9 (ran B dom A → (xByy dom A))
6 eldm 4898 . . . . . . . . 9 (y dom Az yAz)
75, 6syl6ib 217 . . . . . . . 8 (ran B dom A → (xByz yAz))
87ancld 536 . . . . . . 7 (ran B dom A → (xBy → (xBy z yAz)))
9 19.42v 1905 . . . . . . 7 (z(xBy yAz) ↔ (xBy z yAz))
108, 9syl6ibr 218 . . . . . 6 (ran B dom A → (xByz(xBy yAz)))
1110eximdv 1622 . . . . 5 (ran B dom A → (y xByyz(xBy yAz)))
12 brco 4883 . . . . . . 7 (x(A B)zy(xBy yAz))
1312exbii 1582 . . . . . 6 (z x(A B)zzy(xBy yAz))
14 excom 1741 . . . . . 6 (zy(xBy yAz) ↔ yz(xBy yAz))
1513, 14bitri 240 . . . . 5 (z x(A B)zyz(xBy yAz))
1611, 15syl6ibr 218 . . . 4 (ran B dom A → (y xByz x(A B)z))
17 eldm 4898 . . . 4 (x dom By xBy)
18 eldm 4898 . . . 4 (x dom (A B) ↔ z x(A B)z)
1916, 17, 183imtr4g 261 . . 3 (ran B dom A → (x dom Bx dom (A B)))
2019ssrdv 3278 . 2 (ran B dom A → dom B dom (A B))
212, 20eqssd 3289 1 (ran B dom A → dom (A B) = dom B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257   class class class wbr 4639   ∘ ccom 4721  dom cdm 4772  ran crn 4773 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-cnv 4785  df-rn 4786  df-dm 4787 This theorem is referenced by:  dmcoeq  4974  fnco  5191  sbthlem3  6205
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