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Theorem rnco 5897
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 3401 . . . . . 6 𝑥 ∈ V
2 vex 3401 . . . . . 6 𝑦 ∈ V
31, 2brco 5540 . . . . 5 (𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
43exbii 1892 . . . 4 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
5 excom 2155 . . . 4 (∃𝑥𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦))
6 vex 3401 . . . . . . . 8 𝑧 ∈ V
76elrn 5614 . . . . . . 7 (𝑧 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑧)
87anbi1i 617 . . . . . 6 ((𝑧 ∈ ran 𝐵𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦))
92brresi 5653 . . . . . 6 (𝑧(𝐴 ↾ ran 𝐵)𝑦 ↔ (𝑧 ∈ ran 𝐵𝑧𝐴𝑦))
10 19.41v 1992 . . . . . 6 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ (∃𝑥 𝑥𝐵𝑧𝑧𝐴𝑦))
118, 9, 103bitr4ri 296 . . . . 5 (∃𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1211exbii 1892 . . . 4 (∃𝑧𝑥(𝑥𝐵𝑧𝑧𝐴𝑦) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
134, 5, 123bitri 289 . . 3 (∃𝑥 𝑥(𝐴𝐵)𝑦 ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
142elrn 5614 . . 3 (𝑦 ∈ ran (𝐴𝐵) ↔ ∃𝑥 𝑥(𝐴𝐵)𝑦)
152elrn 5614 . . 3 (𝑦 ∈ ran (𝐴 ↾ ran 𝐵) ↔ ∃𝑧 𝑧(𝐴 ↾ ran 𝐵)𝑦)
1613, 14, 153bitr4i 295 . 2 (𝑦 ∈ ran (𝐴𝐵) ↔ 𝑦 ∈ ran (𝐴 ↾ ran 𝐵))
1716eqriv 2775 1 ran (𝐴𝐵) = ran (𝐴 ↾ ran 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1601  wex 1823  wcel 2107   class class class wbr 4888  ran crn 5358  cres 5359  ccom 5361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-xp 5363  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369
This theorem is referenced by:  rnco2  5898  coeq0  5900  cofunexg  7411  1stcof  7477  2ndcof  7478  smobeth  9745  elmsubrn  32032  ftc1anclem3  34121
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