Theorem funco 4862

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 4524	. . . . 5 ⊢ dom (𝐹 ∘ 𝐺) ⊆ dom 𝐺
2		funmo 4839	. . . . . . . . . 10 ⊢ (Fun 𝐹 → ∃*y z𝐹y)
3	2	alrimiv 1732	. . . . . . . . 9 ⊢ (Fun 𝐹 → ∀z∃*y z𝐹y)
4	3	ralrimivw 2367	. . . . . . . 8 ⊢ (Fun 𝐹 → ∀x ∈ dom 𝐺∀z∃*y z𝐹y)
5		dffun8 4851	. . . . . . . . 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 2415	. . . . . . 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 1398	. . . . . . . 8 ⊢ Ⅎy x𝐺z
11	10	euexex 1963	. . . . . . 7 ⊢ ((∃!z x𝐺z ∧ ∀z∃*y z𝐹y) → ∃*y∃z(x𝐺z ∧ z𝐹y))
12	11	ralimi 2358	. . . . . 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 2977	. . . . 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 3735	. . . . . . 7 ⊢ (x(𝐹 ∘ 𝐺)y ↔ ⟨x, y⟩ ∈ (𝐹 ∘ 𝐺))
17		df-co 4277	. . . . . . . 8 ⊢ (𝐹 ∘ 𝐺) = {⟨x, y⟩ ∣ ∃z(x𝐺z ∧ z𝐹y)}
18	17	eleq2i 2082	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (𝐹 ∘ 𝐺) ↔ ⟨x, y⟩ ∈ {⟨x, y⟩ ∣ ∃z(x𝐺z ∧ z𝐹y)})
19		opabid 3964	. . . . . . 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 1915	. . . . 5 ⊢ (∃*y x(𝐹 ∘ 𝐺)y ↔ ∃*y∃z(x𝐺z ∧ z𝐹y))
22	21	ralbii 2304	. . . 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 4742	. . 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 4850	. 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 1224  ∃wex 1358   ∈ wcel 1370  ∃!weu 1878  ∃*wmo 1879  ∀wral 2280   ⊆ wss 2890  ⟨cop 3349   class class class wbr 3734  {copab 3787  dom cdm 4268   ∘ ccom 4272  Rel wrel 4273  Fun wfun 4819

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-fun 4827

This theorem is referenced by:  fnco  4929  f1co  5022  tposfun  5793