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Theorem dmcosseq 4546
Description: Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmcosseq (ran B ⊆ dom A → dom (AB) = dom B)

Proof of Theorem dmcosseq
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmcoss 4544 . . 3 dom (AB) ⊆ dom B
21a1i 9 . 2 (ran B ⊆ dom A → dom (AB) ⊆ dom B)
3 ssel 2933 . . . . . . . 8 (ran B ⊆ dom A → (y ran By dom A))
4 vex 2554 . . . . . . . . . . 11 y V
54elrn 4520 . . . . . . . . . 10 (y ran Bx xBy)
64eldm 4475 . . . . . . . . . 10 (y dom Az yAz)
75, 6imbi12i 228 . . . . . . . . 9 ((y ran By dom A) ↔ (x xByz yAz))
8 19.8a 1479 . . . . . . . . . . 11 (xByx xBy)
98imim1i 54 . . . . . . . . . 10 ((x xByz yAz) → (xByz yAz))
10 pm3.2 126 . . . . . . . . . . 11 (xBy → (yAz → (xBy yAz)))
1110eximdv 1757 . . . . . . . . . 10 (xBy → (z yAzz(xBy yAz)))
129, 11sylcom 25 . . . . . . . . 9 ((x xByz yAz) → (xByz(xBy yAz)))
137, 12sylbi 114 . . . . . . . 8 ((y ran By dom A) → (xByz(xBy yAz)))
143, 13syl 14 . . . . . . 7 (ran B ⊆ dom A → (xByz(xBy yAz)))
1514eximdv 1757 . . . . . 6 (ran B ⊆ dom A → (y xByyz(xBy yAz)))
16 excom 1551 . . . . . 6 (zy(xBy yAz) ↔ yz(xBy yAz))
1715, 16syl6ibr 151 . . . . 5 (ran B ⊆ dom A → (y xByzy(xBy yAz)))
18 vex 2554 . . . . . . 7 x V
19 vex 2554 . . . . . . 7 z V
2018, 19opelco 4450 . . . . . 6 (⟨x, z (AB) ↔ y(xBy yAz))
2120exbii 1493 . . . . 5 (zx, z (AB) ↔ zy(xBy yAz))
2217, 21syl6ibr 151 . . . 4 (ran B ⊆ dom A → (y xByzx, z (AB)))
2318eldm 4475 . . . 4 (x dom By xBy)
2418eldm2 4476 . . . 4 (x dom (AB) ↔ zx, z (AB))
2522, 23, 243imtr4g 194 . . 3 (ran B ⊆ dom A → (x dom Bx dom (AB)))
2625ssrdv 2945 . 2 (ran B ⊆ dom A → dom B ⊆ dom (AB))
272, 26eqssd 2956 1 (ran B ⊆ dom A → dom (AB) = dom B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  wss 2911  cop 3370   class class class wbr 3755  dom cdm 4288  ran crn 4289  ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299
This theorem is referenced by:  dmcoeq  4547  fnco  4950
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