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Theorem funco 6378
 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 6354 . . . . 5 (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧)
2 funmo 6354 . . . . . 6 (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
32alrimiv 1929 . . . . 5 (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4 moexexvw 2716 . . . . 5 ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
51, 3, 4syl2anr 599 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
65alrimiv 1929 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
7 funopab 6373 . . 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 5547 . . 3 (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)}
109funeqi 6359 . 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 1536  ∃wex 1781  ∃*wmo 2622   class class class wbr 5049  {copab 5111   ∘ ccom 5542  Fun wfun 6332 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 5186  ax-nul 5193  ax-pr 5313 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 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-br 5050  df-opab 5112  df-id 5443  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-fun 6340 This theorem is referenced by:  funresfunco  6379  fnco  6448  f1co  6568  curry1  7784  curry2  7787  tposfun  7893  fsuppco  8851  fsuppco2  8852  fsuppcor  8853  fin23lem30  9751  smobeth  9995  hashkf  13688  xppreima  30393  smatrcl  31084  comptiunov2i  40254  fco3  41713  hoicvr  43044
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