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Theorem rnco 5600
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 3189 . . . . . 6 𝑥 ∈ V
2 vex 3189 . . . . . 6 𝑦 ∈ V
31, 2brco 5252 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
43exbii 1771 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 excom 2039 . . . 4 (∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦))
6 ancom 466 . . . . . . 7 ((∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧))
7 19.41v 1911 . . . . . . 7 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦))
8 vex 3189 . . . . . . . . 9 𝑧 ∈ V
98elrn 5326 . . . . . . . 8 (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
109anbi2i 729 . . . . . . 7 ((𝑧𝐴𝑦𝑧 ∈ ran 𝐵) ↔ (𝑧𝐴𝑦 ∧ ∃𝑥 𝑥𝐵𝑧))
116, 7, 103bitr4i 292 . . . . . 6 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (𝑧𝐴𝑦𝑧 ∈ ran 𝐵))
122brres 5362 . . . . . 6 (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧𝐴𝑦𝑧 ∈ ran 𝐵))
1311, 12bitr4i 267 . . . . 5 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1413exbii 1771 . . . 4 (∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
154, 5, 143bitri 286 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
162elrn 5326 . . 3 (𝑦 ∈ ran (𝐴𝐵) ↔ ∃𝑥 𝑥(𝐴𝐵)𝑦)
172elrn 5326 . . 3 (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1815, 16, 173bitr4i 292 . 2 (𝑦 ∈ ran (𝐴𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
1918eqriv 2618 1 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987   class class class wbr 4613  ran crn 5075  cres 5076  ccom 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086
This theorem is referenced by:  rnco2  5601  coeq0  5603  cofunexg  7077  1stcof  7141  2ndcof  7142  smobeth  9352  elmsubrn  31133  ftc1anclem3  33119
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