Theorem funcnv 5156

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 5155 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 4960	. . . . . 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 5155	. 2 ⊢ (Fun ◡A ↔ ∀y∃*x xAy)
8		df-ral 2619	. 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 2614   class class class wbr 4639  ^◡ccnv 4771  ran crn 4773  Fun wfun 4775

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789

This theorem is referenced by:  funcnv3  5157  fncnv  5158