Theorem fununmo 5315

Description: If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.)

Assertion

Ref	Expression
fununmo	|- ( Fun ( F u. G ) -> E* y x F y )

Distinct variable groups:   x, y   y, F   y, G

Allowed substitution hints:   F( x)   G( x)

Proof of Theorem fununmo

Step	Hyp	Ref	Expression
1		funmo 5285	. 2 |- ( Fun ( F u. G ) -> E* y x ( F u. G ) y )
2		orc 713	. . . 4 |- ( x F y -> ( x F y \/ x G y ) )
3		brun 4094	. . . 4 |- ( x ( F u. G ) y <-> ( x F y \/ x G y ) )
4	2, 3	sylibr 134	. . 3 |- ( x F y -> x ( F u. G ) y )
5	4	moimi 2118	. 2 |- ( E* y x ( F u. G ) y -> E* y x F y )
6	1, 5	syl 14	1 |- ( Fun ( F u. G ) -> E* y x F y )

Syntax hints:   -> wi 4   \/ wo 709   E*wmo 2054   u. cun 3163   class class class wbr 4043   Fun wfun 5264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-fun 5272

This theorem is referenced by:  fununfun  5316