Theorem funco 6225

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 6201	. . . . 5 ⊢ (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧)
2		funmo 6201	. . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
3	2	alrimiv 1886	. . . . 5 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4		moexexv 2671	. . . . 5 ⊢ ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
5	1, 3, 4	syl2anr 587	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
6	5	alrimiv 1886	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
7		funopab 6220	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
8	6, 7	sylibr 226	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
9		df-co 5412	. . 3 ⊢ (𝐹 ∘ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}
10	9	funeqi 6206	. 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
11	8, 10	sylibr 226	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Syntax hints:   → wi 4   ∧ wa 387  ∀wal 1505  ∃wex 1742  ∃*wmo 2545   class class class wbr 4925  {copab 4987   ∘ ccom 5407  Fun wfun 6179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-fun 6187

This theorem is referenced by:  funresfunco  6226  fnco  6295  f1co  6411  curry1  7605  curry2  7608  tposfun  7709  fsuppco  8658  fsuppco2  8659  fsuppcor  8660  fin23lem30  9560  smobeth  9804  hashkf  13505  xppreima  30170  smatrcl  30732  comptiunov2i  39443  fco3  40942  hoicvr  42286