Theorem funmo 5126

Description: A function has at most one value for each argument. (Contributed by NM, 24-May-1998.)

Assertion

Ref	Expression
funmo	|- (Fun F -> E*y AFy)

Distinct variable groups:   y,A   y,F

Proof of Theorem funmo

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brreldmex 4691	. . . 4 |- (AFy -> A e. _V)
2	1	ancri 535	. . 3 |- (AFy -> (A e. _V /\ AFy))
3	2	ax-gen 1546	. 2 |- A.y(AFy -> (A e. _V /\ AFy))
4		breq1 4643	. . . . . 6 |- (x = A -> (xFy <-> AFy))
5	4	mobidv 2239	. . . . 5 |- (x = A -> (E*y xFy <-> E*y AFy))
6	5	spcgv 2940	. . . 4 |- (A e. _V -> (A.xE*y xFy -> E*y AFy))
7	6	com12 27	. . 3 |- (A.xE*y xFy -> (A e. _V -> E*y AFy))
8		dffun6 5125	. . 3 |- (Fun F <-> A.xE*y xFy)
9		moanimv 2262	. . 3 |- (E*y(A e. _V /\ AFy) <-> (A e. _V -> E*y AFy))
10	7, 8, 9	3imtr4i 257	. 2 |- (Fun F -> E*y(A e. _V /\ AFy))
11		moim 2250	. 2 |- (A.y(AFy -> (A e. _V /\ AFy)) -> (E*y(A e. _V /\ AFy) -> E*y AFy))
12	3, 10, 11	mpsyl 59	1 |- (Fun F -> E*y AFy)

Syntax hints:   -> wi 4   /\ wa 358  A.wal 1540   = wceq 1642   e. wcel 1710  E*wmo 2205  _Vcvv 2860   class class class wbr 4640  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-id 4768  df-cnv 4786  df-fun 4790

This theorem is referenced by:  funeu  5132  funco  5143  imadif  5172  fneu  5188  dff3  5421