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Theorem rnco 6092
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 3483 . . . . . 6 𝑥 ∈ V
2 vex 3483 . . . . . 6 𝑦 ∈ V
31, 2brco 5728 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
43exbii 1849 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 excom 2170 . . . 4 (∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦))
6 vex 3483 . . . . . . . 8 𝑧 ∈ V
76elrn 5809 . . . . . . 7 (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
87anbi1i 626 . . . . . 6 ((𝑧 ∈ ran 𝐵𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦))
92brresi 5849 . . . . . 6 (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧 ∈ ran 𝐵𝑧𝐴𝑦))
10 19.41v 1951 . . . . . 6 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦))
118, 9, 103bitr4ri 307 . . . . 5 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1211exbii 1849 . . . 4 (∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
134, 5, 123bitri 300 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
142elrn 5809 . . 3 (𝑦 ∈ ran (𝐴𝐵) ↔ ∃𝑥 𝑥(𝐴𝐵)𝑦)
152elrn 5809 . . 3 (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1613, 14, 153bitr4i 306 . 2 (𝑦 ∈ ran (𝐴𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
1716eqriv 2821 1 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wex 1781  wcel 2115   class class class wbr 5052  ran crn 5543  cres 5544  ccom 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554
This theorem is referenced by:  rnco2  6093  coeq0  6095  cofunexg  7640  1stcof  7709  2ndcof  7710  smobeth  10000  cycpmconjv  30809  elmsubrn  32800  ftc1anclem3  35042
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