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Theorem funco 5842
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 5820 . . . . 5 (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧)
2 funmo 5820 . . . . . 6 (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
32alrimiv 1842 . . . . 5 (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4 moexexv 2530 . . . . 5 ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
51, 3, 4syl2anr 494 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
65alrimiv 1842 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
7 funopab 5837 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
86, 7sylibr 223 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
9 df-co 5047 . . 3 (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)}
109funeqi 5824 . 2 (Fun (𝐹𝐺) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
118, 10sylibr 223 1 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1473  wex 1695  ∃*wmo 2459   class class class wbr 4583  {copab 4642  ccom 5042  Fun wfun 5798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806
This theorem is referenced by:  fnco  5913  f1co  6023  curry1  7156  curry2  7159  tposfun  7255  fsuppco  8190  fsuppco2  8191  fsuppcor  8192  fin23lem30  9047  smobeth  9287  hashkf  12981  xppreima  28829  smatrcl  29190  comptiunov2i  37017  fco3  38416  hoicvr  39438  funresfunco  39854
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