Theorem dffun9 5362

Description: Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
dffun9	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dffun9

Step	Hyp	Ref	Expression
1		dffun7 5360	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
2		vex 2806	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 2806	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	brelrn 4971	. . . . . . 7 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
5	4	pm4.71ri 392	. . . . . 6 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6	5	mobii 2116	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
7		df-rmo 2519	. . . . 5 ⊢ (∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
8	6, 7	bitr4i 187	. . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)
9	8	ralbii 2539	. . 3 ⊢ (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)
10	9	anbi2i 457	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))
11	1, 10	bitri 184	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃*wmo 2080   ∈ wcel 2202  ∀wral 2511  ∃*wrmo 2514   class class class wbr 4093  dom cdm 4731  ran crn 4732  Rel wrel 4736  Fun wfun 5327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rmo 2519  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-fun 5335

This theorem is referenced by: (None)