Theorem funco 4881

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 4543	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
2		funmo 4858	. . . . . . . . . 10 ⊢ (Fun 𝐹 → ∃*y z𝐹y)
3	2	alrimiv 1751	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀z∃*y z𝐹y)
4	3	ralrimivw 2387	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀x ∈ dom 𝐺∀z∃*y z𝐹y)
5		dffun8 4870	. . . . . . . . 9 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀x ∈ dom 𝐺∃!z x𝐺z))
6	5	simprbi 260	. . . . . . . 8 ⊢ (Fun 𝐺 → ∀x ∈ dom 𝐺∃!z x𝐺z)
7	4, 6	anim12ci 322	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (∀x ∈ dom 𝐺∃!z x𝐺z ∧ ∀x ∈ dom 𝐺∀z∃*y z𝐹y))
8		r19.26 2435	. . . . . . 7 ⊢ (∀x ∈ dom 𝐺(∃!z x𝐺z ∧ ∀z∃*y z𝐹y) ↔ (∀x ∈ dom 𝐺∃!z x𝐺z ∧ ∀x ∈ dom 𝐺∀z∃*y z𝐹y))
9	7, 8	sylibr 137	. . . . . 6 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀x ∈ dom 𝐺(∃!z x𝐺z ∧ ∀z∃*y z𝐹y))
10		nfv 1418	. . . . . . . 8 ⊢ Ⅎy x𝐺z
11	10	euexex 1982	. . . . . . 7 ⊢ ((∃!z x𝐺z ∧ ∀z∃*y z𝐹y) → ∃*y∃z(x𝐺z ∧ z𝐹y))
12	11	ralimi 2378	. . . . . 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 2998	. . . . 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 59	. . . 4 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀x ∈ dom (𝐹 ∘ 𝐺)∃*y∃z(x𝐺z ∧ z𝐹y))
16		df-br 3755	. . . . . . 7 ⊢ (x(𝐹 ∘ 𝐺)y ↔ ⟨x, y⟩ ∈ (𝐹 ∘ 𝐺))
17		df-co 4296	. . . . . . . 8 ⊢ (𝐹 ∘ 𝐺) = {⟨x, y⟩ ∣ ∃z(x𝐺z ∧ z𝐹y)}
18	17	eleq2i 2101	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (𝐹 ∘ 𝐺) ↔ ⟨x, y⟩ ∈ {⟨x, y⟩ ∣ ∃z(x𝐺z ∧ z𝐹y)})
19		opabid 3984	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩ ∣ ∃z(x𝐺z ∧ z𝐹y)} ↔ ∃z(x𝐺z ∧ z𝐹y))
20	16, 18, 19	3bitri 195	. . . . . 6 ⊢ (x(𝐹 ∘ 𝐺)y ↔ ∃z(x𝐺z ∧ z𝐹y))
21	20	mobii 1934	. . . . 5 ⊢ (∃*y x(𝐹 ∘ 𝐺)y ↔ ∃*y∃z(x𝐺z ∧ z𝐹y))
22	21	ralbii 2324	. . . 4 ⊢ (∀x ∈ dom (𝐹 ∘ 𝐺)∃*y x(𝐹 ∘ 𝐺)y ↔ ∀x ∈ dom (𝐹 ∘ 𝐺)∃*y∃z(x𝐺z ∧ z𝐹y))
23	15, 22	sylibr 137	. . 3 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → ∀x ∈ dom (𝐹 ∘ 𝐺)∃*y x(𝐹 ∘ 𝐺)y)
24		relco 4761	. . 3 ⊢ Rel (𝐹 ∘ 𝐺)
25	23, 24	jctil 295	. 2 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (Rel (𝐹 ∘ 𝐺) ∧ ∀x ∈ dom (𝐹 ∘ 𝐺)∃*y x(𝐹 ∘ 𝐺)y))
26		dffun7 4869	. 2 ⊢ (Fun (𝐹 ∘ 𝐺) ↔ (Rel (𝐹 ∘ 𝐺) ∧ ∀x ∈ dom (𝐹 ∘ 𝐺)∃*y x(𝐹 ∘ 𝐺)y))
27	25, 26	sylibr 137	1 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  ∃*wmo 1898  ∀wral 2300   ⊆ wss 2911  ⟨cop 3369   class class class wbr 3754  {copab 3807  dom cdm 4287   ∘ ccom 4291  Rel wrel 4292  Fun wfun 4838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-fun 4846

This theorem is referenced by:  fnco  4948  f1co  5042  tposfun  5813