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Theorem funco 6429
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 6405 . . . . 5 (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧)
2 funmo 6405 . . . . . 6 (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
32alrimiv 1935 . . . . 5 (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4 moexexvw 2630 . . . . 5 ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
51, 3, 4syl2anr 600 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
65alrimiv 1935 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
7 funopab 6424 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
86, 7sylibr 237 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
9 df-co 5569 . . 3 (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)}
109funeqi 6410 . 2 (Fun (𝐹𝐺) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
118, 10sylibr 237 1 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1541  wex 1787  ∃*wmo 2538   class class class wbr 5062  {copab 5124  ccom 5564  Fun wfun 6383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pr 5331
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-sn 4551  df-pr 4553  df-op 4557  df-br 5063  df-opab 5125  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-fun 6391
This theorem is referenced by:  funresfunco  6430  fncofn  6502  fncoOLD  6504  fco3OLD  6588  f1cof1  6635  f1coOLD  6637  curry1  7881  curry2  7884  tposfun  7993  fsuppco  9031  fsuppco2  9032  fsuppcor  9033  fin23lem30  9969  smobeth  10213  hashkf  13911  xppreima  30715  smatrcl  31473  comptiunov2i  41006  hoicvr  43776
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