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Theorem funco 6071
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 6047 . . . . 5 (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧)
2 funmo 6047 . . . . . 6 (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
32alrimiv 2007 . . . . 5 (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4 moexexv 2691 . . . . 5 ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
51, 3, 4syl2anr 584 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
65alrimiv 2007 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
7 funopab 6066 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦𝑧(𝑥𝐺𝑧𝑧𝐹𝑦))
86, 7sylibr 224 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
9 df-co 5258 . . 3 (𝐹𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)}
109funeqi 6052 . 2 (Fun (𝐹𝐺) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧𝑧𝐹𝑦)})
118, 10sylibr 224 1 ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629  wex 1852  ∃*wmo 2619   class class class wbr 4786  {copab 4846  ccom 5253  Fun wfun 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-fun 6033
This theorem is referenced by:  fnco  6139  f1co  6251  curry1  7420  curry2  7423  tposfun  7520  fsuppco  8463  fsuppco2  8464  fsuppcor  8465  fin23lem30  9366  smobeth  9610  hashkf  13323  xppreima  29789  smatrcl  30202  comptiunov2i  38524  fco3  39939  hoicvr  41282  funresfunco  41725
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