Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  rnco GIF version

Theorem rnco 4827
 Description: The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.)
Assertion
Ref Expression
rnco ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)

Proof of Theorem rnco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . . . 6 𝑥 ∈ V
2 vex 2560 . . . . . 6 𝑦 ∈ V
31, 2brco 4506 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
43exbii 1496 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 excom 1554 . . . 4 (∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦))
6 ancom 253 . . . . . . 7 ((∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧))
7 19.41v 1782 . . . . . . 7 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦))
8 vex 2560 . . . . . . . . 9 𝑧 ∈ V
98elrn 4577 . . . . . . . 8 (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
109anbi2i 430 . . . . . . 7 ((𝑧𝐴𝑦𝑧 ∈ ran 𝐵) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧))
116, 7, 103bitr4i 201 . . . . . 6 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦𝑧 ∈ ran 𝐵))
122brres 4618 . . . . . 6 (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧𝐴𝑦𝑧 ∈ ran 𝐵))
1311, 12bitr4i 176 . . . . 5 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1413exbii 1496 . . . 4 (∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
154, 5, 143bitri 195 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
162elrn 4577 . . 3 (𝑦 ∈ ran (𝐴𝐵) ↔ ∃𝑥 𝑥(𝐴𝐵)𝑦)
172elrn 4577 . . 3 (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1815, 16, 173bitr4i 201 . 2 (𝑦 ∈ ran (𝐴𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
1918eqriv 2037 1 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393   class class class wbr 3764  ran crn 4346   ↾ cres 4347   ∘ ccom 4349 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357 This theorem is referenced by:  rnco2  4828  cofunexg  5738  1stcof  5790  2ndcof  5791
 Copyright terms: Public domain W3C validator