Theorem funin 2708

Description: The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53.

Assertion

Ref	Expression
funin	⊢ (Fun F → Fun (F ∩ G))

Proof of Theorem funin

Step	Hyp	Ref	Expression
1		relin 2491	. . 3 ⊢ (Rel F → Rel (F ∩ G))
2		moan 1046	. . . . 5 ⊢ (∃*y xFy → ∃*y(⟨x, y⟩ ∈ G ∧ xFy))
3		ancom 333	. . . . . . 7 ⊢ ((⟨x, y⟩ ∈ G ∧ xFy) ↔ (xFy ∧ ⟨x, y⟩ ∈ G))
4		elin 1635	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ (F ∩ G) ↔ (⟨x, y⟩ ∈ F ∧ ⟨x, y⟩ ∈ G))
5		df-br 2063	. . . . . . . 8 ⊢ (x(F ∩ G)y ↔ ⟨x, y⟩ ∈ (F ∩ G))
6		df-br 2063	. . . . . . . . 9 ⊢ (xFy ↔ ⟨x, y⟩ ∈ F)
7	6	anbi1i 368	. . . . . . . 8 ⊢ ((xFy ∧ ⟨x, y⟩ ∈ G) ↔ (⟨x, y⟩ ∈ F ∧ ⟨x, y⟩ ∈ G))
8	4, 5, 7	3bitr4 158	. . . . . . 7 ⊢ (x(F ∩ G)y ↔ (xFy ∧ ⟨x, y⟩ ∈ G))
9	3, 8	bitr4 154	. . . . . 6 ⊢ ((⟨x, y⟩ ∈ G ∧ xFy) ↔ x(F ∩ G)y)
10	9	bimo 1031	. . . . 5 ⊢ (∃*y(⟨x, y⟩ ∈ G ∧ xFy) ↔ ∃*y x(F ∩ G)y)
11	2, 10	sylib 173	. . . 4 ⊢ (∃*y xFy → ∃*y x(F ∩ G)y)
12	11	19.20i 691	. . 3 ⊢ (∀x∃*y xFy → ∀x∃*y x(F ∩ G)y)
13	1, 12	anim12i 268	. 2 ⊢ ((Rel F ∧ ∀x∃*y xFy) → (Rel (F ∩ G) ∧ ∀x∃*y x(F ∩ G)y))
14		dffunmo 2679	. 2 ⊢ (Fun F ↔ (Rel F ∧ ∀x∃*y xFy))
15		dffunmo 2679	. 2 ⊢ (Fun (F ∩ G) ↔ (Rel (F ∩ G) ∧ ∀x∃*y x(F ∩ G)y))
16	13, 14, 15	3imtr4 192	1 ⊢ (Fun F → Fun (F ∩ G))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃*wmo 1008   ∈ wcel 1092   ∩ cin 1486  ⟨cop 1810   class class class wbr 2054  Rel wrel 2415  Fun wfun 2416

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-rel 2425  df-cnv 2426  df-co 2427  df-fun 2432

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