Theorem funco 2696

Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
funco	⊢ ((Fun F ∧ Fun G) → Fun (F ∘ G))

Proof of Theorem funco

Step	Hyp	Ref	Expression
1		moexexv 1059	. . . . . . 7 ⊢ ((∃*z xGz ∧ ∀z∃*y zFy) → ∃*y∃z(xGz ∧ zFy))
2		funmo 2680	. . . . . . 7 ⊢ (Fun G → ∃*z xGz)
3		dffunmo 2679	. . . . . . . 8 ⊢ (Fun F ↔ (Rel F ∧ ∀z∃*y zFy))
4	3	pm3.27bd 263	. . . . . . 7 ⊢ (Fun F → ∀z∃*y zFy)
5	1, 2, 4	syl2an 349	. . . . . 6 ⊢ ((Fun G ∧ Fun F) → ∃*y∃z(xGz ∧ zFy))
6	5	ancoms 334	. . . . 5 ⊢ ((Fun F ∧ Fun G) → ∃*y∃z(xGz ∧ zFy))
7		visset 1350	. . . . . . 7 ⊢ x ∈ V
8		visset 1350	. . . . . . 7 ⊢ y ∈ V
9	7, 8	brco 2510	. . . . . 6 ⊢ (x(F ∘ G)y ↔ ∃z(xGz ∧ zFy))
10	9	bimo 1031	. . . . 5 ⊢ (∃*y x(F ∘ G)y ↔ ∃*y∃z(xGz ∧ zFy))
11	6, 10	sylibr 175	. . . 4 ⊢ ((Fun F ∧ Fun G) → ∃*y x(F ∘ G)y)
12	11	19.21aiv 943	. . 3 ⊢ ((Fun F ∧ Fun G) → ∀x∃*y x(F ∘ G)y)
13		relco 2658	. . 3 ⊢ Rel (F ∘ G)
14	12, 13	jctil 240	. 2 ⊢ ((Fun F ∧ Fun G) → (Rel (F ∘ G) ∧ ∀x∃*y x(F ∘ G)y))
15		dffunmo 2679	. 2 ⊢ (Fun (F ∘ G) ↔ (Rel (F ∘ G) ∧ ∀x∃*y x(F ∘ G)y))
16	14, 15	sylibr 175	1 ⊢ ((Fun F ∧ Fun G) → Fun (F ∘ G))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678  ∃*wmo 1008   class class class wbr 2054   ∘ ccom 2414  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  fco 2760  f1co 2783  f1oco 2816  fvco 2865  mapenlem1 3384

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-fun 2432

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