Theorem funco 5928

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 (𝐹 ∘ 𝐺))

Proof of Theorem funco

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmo 5904	. . . . 5 ⊢ (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧)
2		funmo 5904	. . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
3	2	alrimiv 1855	. . . . 5 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4		moexexv 2542	. . . . 5 ⊢ ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
5	1, 3, 4	syl2anr 495	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
6	5	alrimiv 1855	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
7		funopab 5923	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
8	6, 7	sylibr 224	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
9		df-co 5123	. . 3 ⊢ (𝐹 ∘ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}
10	9	funeqi 5909	. 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
11	8, 10	sylibr 224	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481  ∃wex 1704  ∃*wmo 2471   class class class wbr 4653  {copab 4712   ∘ ccom 5118  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-fun 5890

This theorem is referenced by:  fnco  5999  f1co  6110  curry1  7269  curry2  7272  tposfun  7368  fsuppco  8307  fsuppco2  8308  fsuppcor  8309  fin23lem30  9164  smobeth  9408  hashkf  13119  xppreima  29449  smatrcl  29862  comptiunov2i  37998  fco3  39421  hoicvr  40762  funresfunco  41205