Theorem dffun6 2687

Description: Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one". However, dffun7 2688 shows that it doesn't matter which meaning we pick.)

Assertion

Ref	Expression
dffun6	⊢ (Fun A ↔ (Rel A ∧ ∀x ∈ dom A∃*y xAy))

Distinct variable group(s):   x,y,A

Proof of Theorem dffun6

Step	Hyp	Ref	Expression
1		dffunmo 2679	. 2 ⊢ (Fun A ↔ (Rel A ∧ ∀x∃*y xAy))
2		moabs 1041	. . . . . 6 ⊢ (∃*y xAy ↔ (∃y xAy → ∃*y xAy))
3		visset 1350	. . . . . . . 8 ⊢ x ∈ V
4	3	eldm 2527	. . . . . . 7 ⊢ (x ∈ dom A ↔ ∃y xAy)
5	4	imbi1i 161	. . . . . 6 ⊢ ((x ∈ dom A → ∃*y xAy) ↔ (∃y xAy → ∃*y xAy))
6	2, 5	bitr4 154	. . . . 5 ⊢ (∃*y xAy ↔ (x ∈ dom A → ∃*y xAy))
7	6	bial 695	. . . 4 ⊢ (∀x∃*y xAy ↔ ∀x(x ∈ dom A → ∃*y xAy))
8		df-ral 1205	. . . 4 ⊢ (∀x ∈ dom A∃*y xAy ↔ ∀x(x ∈ dom A → ∃*y xAy))
9	7, 8	bitr4 154	. . 3 ⊢ (∀x∃*y xAy ↔ ∀x ∈ dom A∃*y xAy)
10	9	anbi2i 367	. 2 ⊢ ((Rel A ∧ ∀x∃*y xAy) ↔ (Rel A ∧ ∀x ∈ dom A∃*y xAy))
11	1, 10	bitr 151	1 ⊢ (Fun A ↔ (Rel A ∧ ∀x ∈ dom A∃*y xAy))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678  ∃*wmo 1008   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054  dom cdm 2410  Rel wrel 2415  Fun wfun 2416

This theorem is referenced by:  dffun7 2688

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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