Theorem dffun9 6447

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 6445	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
2		vex 3426	. . . . . . . 8 ⊢ 𝑥 ∈ V
3		vex 3426	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	2, 3	brelrn 5840	. . . . . . 7 ⊢ (𝑥𝐴𝑦 → 𝑦 ∈ ran 𝐴)
5	4	pm4.71ri 560	. . . . . 6 ⊢ (𝑥𝐴𝑦 ↔ (𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
6	5	mobii 2548	. . . . 5 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
7		df-rmo 3071	. . . . 5 ⊢ (∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦 ↔ ∃*𝑦(𝑦 ∈ ran 𝐴 ∧ 𝑥𝐴𝑦))
8	6, 7	bitr4i 277	. . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)
9	8	ralbii 3090	. . 3 ⊢ (∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)
10	9	anbi2i 622	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))
11	1, 10	bitri 274	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))

Syntax hints:   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∃*wmo 2538  ∀wral 3063  ∃*wrmo 3066   class class class wbr 5070  dom cdm 5580  ran crn 5581  Rel wrel 5585  Fun wfun 6412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rmo 3071  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-fun 6420

This theorem is referenced by:  brdom4  10217