Theorem funco 5294

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		dmcoss 4931	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
2		funmo 5269	. . . . . . . . . 10 ⊢ (Fun 𝐹 → ∃*𝑦 𝑧𝐹𝑦)
3	2	alrimiv 1885	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀𝑧∃*𝑦 𝑧𝐹𝑦)
4	3	ralrimivw 2568	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦)
5		dffun8 5282	. . . . . . . . 9 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧))
6	5	simprbi 275	. . . . . . . 8 ⊢ (Fun 𝐺 → ∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧)
7	4, 6	anim12ci 339	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦))
8		r19.26 2620	. . . . . . 7 ⊢ (∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) ↔ (∀𝑥 ∈ dom 𝐺∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑥 ∈ dom 𝐺∀𝑧∃*𝑦 𝑧𝐹𝑦))
9	7, 8	sylibr 134	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦))
10		nfv 1539	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑥𝐺𝑧
11	10	euexex 2127	. . . . . . 7 ⊢ ((∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
12	11	ralimi 2557	. . . . . 6 ⊢ (∀𝑥 ∈ dom 𝐺(∃!𝑧 𝑥𝐺𝑧 ∧ ∀𝑧∃*𝑦 𝑧𝐹𝑦) → ∀𝑥 ∈ dom 𝐺∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
13	9, 12	syl 14	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom 𝐺∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
14		ssralv 3243	. . . . 5 ⊢ (dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺 → (∀𝑥 ∈ dom 𝐺∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦) → ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)))
15	1, 13, 14	mpsyl 65	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
16		df-br 4030	. . . . . . 7 ⊢ (𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺))
17		df-co 4668	. . . . . . . 8 ⊢ (𝐹 ∘ 𝐺) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)}
18	17	eleq2i 2260	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)})
19		opabid 4286	. . . . . . 7 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦)} ↔ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
20	16, 18, 19	3bitri 206	. . . . . 6 ⊢ (𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
21	20	mobii 2079	. . . . 5 ⊢ (∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
22	21	ralbii 2500	. . . 4 ⊢ (∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦 ↔ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦∃𝑧(𝑥𝐺𝑧 ∧ 𝑧𝐹𝑦))
23	15, 22	sylibr 134	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦)
24		relco 5164	. . 3 ⊢ Rel (𝐹 ∘ 𝐺)
25	23, 24	jctil 312	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (Rel (𝐹 ∘ 𝐺) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦))
26		dffun7 5281	. 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ (Rel (𝐹 ∘ 𝐺) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∃*𝑦 𝑥(𝐹 ∘ 𝐺)𝑦))
27	25, 26	sylibr 134	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1362  ∃wex 1503  ∃!weu 2042  ∃*wmo 2043   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3153  ⟨cop 3621   class class class wbr 4029  {copab 4089  dom cdm 4659   ∘ ccom 4663  Rel wrel 4664  Fun wfun 5248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-fun 5256

This theorem is referenced by:  fnco  5362  f1co  5471  tposfun  6313  casefun  7144  caseinj  7148  caseinl  7150  caseinr  7151  djufun  7163  djuinj  7165  ctssdccl  7170  lidlmex  13971