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Theorem rnco 5115
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 2733 . . . . . 6 𝑥 ∈ V
2 vex 2733 . . . . . 6 𝑦 ∈ V
31, 2brco 4780 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
43exbii 1598 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 excom 1657 . . . 4 (∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦))
6 ancom 264 . . . . . . 7 ((∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧))
7 19.41v 1895 . . . . . . 7 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦))
8 vex 2733 . . . . . . . . 9 𝑧 ∈ V
98elrn 4852 . . . . . . . 8 (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
109anbi2i 454 . . . . . . 7 ((𝑧𝐴𝑦𝑧 ∈ ran 𝐵) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧))
116, 7, 103bitr4i 211 . . . . . 6 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦𝑧 ∈ ran 𝐵))
122brres 4895 . . . . . 6 (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧𝐴𝑦𝑧 ∈ ran 𝐵))
1311, 12bitr4i 186 . . . . 5 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1413exbii 1598 . . . 4 (∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
154, 5, 143bitri 205 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
162elrn 4852 . . 3 (𝑦 ∈ ran (𝐴𝐵) ↔ ∃𝑥 𝑥(𝐴𝐵)𝑦)
172elrn 4852 . . 3 (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1815, 16, 173bitr4i 211 . 2 (𝑦 ∈ ran (𝐴𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
1918eqriv 2167 1 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1348  wex 1485  wcel 2141   class class class wbr 3987  ran crn 4610  cres 4611  ccom 4613
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621
This theorem is referenced by:  rnco2  5116  cofunexg  6085  1stcof  6139  2ndcof  6140  djudom  7066
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