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Theorem rnco 6274
Description: The range of the composition of two classes. (Contributed by NM, 12-Dec-2006.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
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 3482 . . . . . 6 𝑥 ∈ V
2 vex 3482 . . . . . 6 𝑦 ∈ V
31, 2brco 5884 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
43exbii 1845 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 excom 2160 . . . 4 (∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦))
6 vex 3482 . . . . . . . 8 𝑧 ∈ V
76elrn 5907 . . . . . . 7 (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
87anbi1i 624 . . . . . 6 ((𝑧 ∈ ran 𝐵𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦))
92brresi 6009 . . . . . 6 (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧 ∈ ran 𝐵𝑧𝐴𝑦))
10 19.41v 1947 . . . . . 6 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦))
118, 9, 103bitr4ri 304 . . . . 5 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1211exbii 1845 . . . 4 (∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
134, 5, 123bitri 297 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
142elrn 5907 . . 3 (𝑦 ∈ ran (𝐴𝐵) ↔ ∃𝑥 𝑥(𝐴𝐵)𝑦)
152elrn 5907 . . 3 (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1613, 14, 153bitr4i 303 . 2 (𝑦 ∈ ran (𝐴𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
1716eqriv 2732 1 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wex 1776  wcel 2106   class class class wbr 5148  ran crn 5690  cres 5691  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701
This theorem is referenced by:  rnco2  6275  coeq0  6277  focofo  6834  cofunexg  7972  1stcof  8043  2ndcof  8044  smobeth  10624  cycpmconjv  33145  elmsubrn  35513  ftc1anclem3  37682
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