Theorem funeu 5913

Description: There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
funeu	|- ( ( Fun F /\ A F B ) -> E! y A F y )

Distinct variable groups:   y, A   y, F

Allowed substitution hint:   B( y)

Proof of Theorem funeu

Step	Hyp	Ref	Expression
1		funrel 5905	. . . 4 |- ( Fun F -> Rel F )
2		releldm 5358	. . . 4 |- ( ( Rel F /\ A F B ) -> A e. dom F )
3	1, 2	sylan 488	. . 3 |- ( ( Fun F /\ A F B ) -> A e. dom F )
4		eldmg 5319	. . . 4 |- ( A e. dom F -> ( A e. dom F <-> E. y A F y ) )
5	4	ibi 256	. . 3 |- ( A e. dom F -> E. y A F y )
6	3, 5	syl 17	. 2 |- ( ( Fun F /\ A F B ) -> E. y A F y )
7		funmo 5904	. . . 4 |- ( Fun F -> E* y A F y )
8	7	adantr 481	. . 3 |- ( ( Fun F /\ A F B ) -> E* y A F y )
9		df-mo 2475	. . 3 |- ( E* y A F y <-> ( E. y A F y -> E! y A F y ) )
10	8, 9	sylib 208	. 2 |- ( ( Fun F /\ A F B ) -> ( E. y A F y -> E! y A F y ) )
11	6, 10	mpd 15	1 |- ( ( Fun F /\ A F B ) -> E! y A F y )

Syntax hints:   -> wi 4   /\ wa 384   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   class class class wbr 4653   dom cdm 5114   Rel wrel 5119   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890

This theorem is referenced by:  funeu2  5914  funbrfv  6234  frege124d  38053