Theorem funco 6526

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 6502	. . . . 5 ⊢ (Fun 𝐺 → ∃*𝑧 𝑥𝐺𝑧)
2		funmo 6502	. . . . . 6 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
3	2	alrimiv 1927	. . . . 5 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4		moexexvw 2621	. . . . 5 ⊢ ((∃*𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
5	1, 3, 4	syl2anr 597	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
6	5	alrimiv 1927	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
7		funopab 6521	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∀𝑥∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
8	6, 7	sylibr 234	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
9		df-co 5632	. . 3 ⊢ (𝐹 ∘ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}
10	9	funeqi 6507	. 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
11	8, 10	sylibr 234	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538  ∃wex 1779  ∃*wmo 2531   class class class wbr 5095  {copab 5157   ∘ ccom 5627  Fun wfun 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-fun 6488

This theorem is referenced by:  funresfunco  6527  fncofn  6603  f1cof1  6734  curry1  8044  curry2  8047  tposfun  8182  fsuppco  9311  fsuppco2  9312  fsuppcor  9313  fin23lem30  10255  smobeth  10499  hashkf  14257  precsexlem10  28141  precsexlem11  28142  xppreima  32602  smatrcl  33762  comptiunov2i  43679  hoicvr  46530  upgrimpthslem1  47892  upgrimspths  47895