Theorem fununmo 5369

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 5339	. 2 |- ( Fun ( F u. G ) -> E* y x ( F u. G ) y )
2		orc 717	. . . 4 |- ( x F y -> ( x F y \/ x G y ) )
3		brun 4138	. . . 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 2143	. 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 713   E*wmo 2078   u. cun 3196   class class class wbr 4086   Fun wfun 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-fun 5326

This theorem is referenced by:  fununfun  5370