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Theorem funco 6388
Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
funco ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))

Proof of Theorem funco
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmo 6364 . . . . 5 (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧)
2 funmo 6364 . . . . . 6 (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
32alrimiv 1919 . . . . 5 (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4 moexexvw 2706 . . . . 5 ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
51, 3, 4syl2anr 596 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
65alrimiv 1919 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
7 funopab 6383 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
86, 7sylibr 235 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
9 df-co 5557 . . 3 (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)}
109funeqi 6369 . 2 (Fun (𝐹𝐺) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
118, 10sylibr 235 1 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526  wex 1771  ∃*wmo 2613   class class class wbr 5057  {copab 5119  ccom 5552  Fun wfun 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-fun 6350
This theorem is referenced by:  funresfunco  6389  fnco  6458  f1co  6578  curry1  7788  curry2  7791  tposfun  7897  fsuppco  8853  fsuppco2  8854  fsuppcor  8855  fin23lem30  9752  smobeth  9996  hashkf  13680  xppreima  30322  smatrcl  30960  comptiunov2i  39929  fco3  41367  hoicvr  42707
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