Theorem funcnv 5157

Description: The converse of a class is a function iff the class is single-rooted, which means that for any y in the range of A there is at most one x such that xAy. Definition of single-rooted in [Enderton] p. 43. See funcnv2 5156 for a simpler version. (Contributed by set.mm contributors, 13-Aug-2004.)

Assertion

Ref	Expression
funcnv	⊢ (Fun ◡A ↔ ∀y ∈ ran A∃*x xAy)

Distinct variable group:   x,y,A

Proof of Theorem funcnv

Step	Hyp	Ref	Expression
1		brelrn 4961	. . . . . 6 ⊢ (xAy → y ∈ ran A)
2	1	pm4.71ri 614	. . . . 5 ⊢ (xAy ↔ (y ∈ ran A ∧ xAy))
3	2	mobii 2240	. . . 4 ⊢ (∃*x xAy ↔ ∃*x(y ∈ ran A ∧ xAy))
4		moanimv 2262	. . . 4 ⊢ (∃*x(y ∈ ran A ∧ xAy) ↔ (y ∈ ran A → ∃*x xAy))
5	3, 4	bitri 240	. . 3 ⊢ (∃*x xAy ↔ (y ∈ ran A → ∃*x xAy))
6	5	albii 1566	. 2 ⊢ (∀y∃*x xAy ↔ ∀y(y ∈ ran A → ∃*x xAy))
7		funcnv2 5156	. 2 ⊢ (Fun ◡A ↔ ∀y∃*x xAy)
8		df-ral 2620	. 2 ⊢ (∀y ∈ ran A∃*x xAy ↔ ∀y(y ∈ ran A → ∃*x xAy))
9	6, 7, 8	3bitr4i 268	1 ⊢ (Fun ◡A ↔ ∀y ∈ ran A∃*x xAy)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  ∃*wmo 2205  ∀wral 2615   class class class wbr 4640  ^◡ccnv 4772  ran crn 4774  Fun wfun 4776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790

This theorem is referenced by:  funcnv3  5158  fncnv  5159