Theorem dffun7 2688

Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun6 2687.

Assertion

Ref	Expression
dffun7	⊢ (Fun A ↔ (Rel A ∧ ∀x ∈ dom A∃!y xAy))

Distinct variable group(s):   x,y,A

Proof of Theorem dffun7

Step	Hyp	Ref	Expression
1		funrel 2681	. . 3 ⊢ (Fun A → Rel A)
2		ax-17 925	. . . . . 6 ⊢ (Fun A → ∀yFun A)
3		hbeu1 1015	. . . . . 6 ⊢ (∃!y⟨x, y⟩ ∈ A → ∀y∃!y⟨x, y⟩ ∈ A)
4		funeu2 2686	. . . . . . 7 ⊢ ((Fun A ∧ ⟨x, y⟩ ∈ A) → ∃!y⟨x, y⟩ ∈ A)
5	4	exp 291	. . . . . 6 ⊢ (Fun A → (⟨x, y⟩ ∈ A → ∃!y⟨x, y⟩ ∈ A))
6	2, 3, 5	19.23ad 748	. . . . 5 ⊢ (Fun A → (∃y⟨x, y⟩ ∈ A → ∃!y⟨x, y⟩ ∈ A))
7		visset 1350	. . . . . 6 ⊢ x ∈ V
8	7	eldm2 2528	. . . . 5 ⊢ (x ∈ dom A ↔ ∃y⟨x, y⟩ ∈ A)
9		df-br 2063	. . . . . 6 ⊢ (xAy ↔ ⟨x, y⟩ ∈ A)
10	9	bieu 1014	. . . . 5 ⊢ (∃!y xAy ↔ ∃!y⟨x, y⟩ ∈ A)
11	6, 8, 10	3imtr4g 426	. . . 4 ⊢ (Fun A → (x ∈ dom A → ∃!y xAy))
12	11	r19.21aiv 1259	. . 3 ⊢ (Fun A → ∀x ∈ dom A∃!y xAy)
13	1, 12	jca 236	. 2 ⊢ (Fun A → (Rel A ∧ ∀x ∈ dom A∃!y xAy))
14		eumo 1037	. . . . 5 ⊢ (∃!y xAy → ∃*y xAy)
15	14	r19.20si 1254	. . . 4 ⊢ (∀x ∈ dom A∃!y xAy → ∀x ∈ dom A∃*y xAy)
16	15	anim2i 270	. . 3 ⊢ ((Rel A ∧ ∀x ∈ dom A∃!y xAy) → (Rel A ∧ ∀x ∈ dom A∃*y xAy))
17		dffun6 2687	. . 3 ⊢ (Fun A ↔ (Rel A ∧ ∀x ∈ dom A∃*y xAy))
18	16, 17	sylibr 175	. 2 ⊢ ((Rel A ∧ ∀x ∈ dom A∃!y xAy) → Fun A)
19	13, 18	impbi 139	1 ⊢ (Fun A ↔ (Rel A ∧ ∀x ∈ dom A∃!y xAy))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678  ∃!weu 1007  ∃*wmo 1008   ∈ wcel 1092  ∀wral 1201  ⟨cop 1810   class class class wbr 2054  dom cdm 2410  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-ral 1205  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-cnv 2426  df-co 2427  df-dm 2428  df-fun 2432

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