Theorem fnunsn 5430

Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
fnunop.x	|- ( ph -> X e. _V )
fnunop.y	|- ( ph -> Y e. _V )
fnunop.f	|- ( ph -> F Fn D )
fnunop.g	|- G = ( F u. { <. X , Y >. } )
fnunop.e	|- E = ( D u. { X } )
fnunop.d	|- ( ph -> -. X e. D )

Assertion

Ref	Expression
fnunsn	|- ( ph -> G Fn E )

Proof of Theorem fnunsn

Step	Hyp	Ref	Expression
1		fnunop.f	. . 3 |- ( ph -> F Fn D )
2		fnunop.x	. . . 4 |- ( ph -> X e. _V )
3		fnunop.y	. . . 4 |- ( ph -> Y e. _V )
4		fnsng 5368	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> { <. X , Y >. } Fn { X } )
5	2, 3, 4	syl2anc 411	. . 3 |- ( ph -> { <. X , Y >. } Fn { X } )
6		fnunop.d	. . . 4 |- ( ph -> -. X e. D )
7		disjsn 3728	. . . 4 |- ( ( D i^i { X } ) = (/) <-> -. X e. D )
8	6, 7	sylibr 134	. . 3 |- ( ph -> ( D i^i { X } ) = (/) )
9		fnun 5429	. . 3 |- ( ( ( F Fn D /\ { <. X , Y >. } Fn { X } ) /\ ( D i^i { X } ) = (/) ) -> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
10	1, 5, 8, 9	syl21anc 1270	. 2 |- ( ph -> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
11		fnunop.g	. . . 4 |- G = ( F u. { <. X , Y >. } )
12	11	fneq1i 5415	. . 3 |- ( G Fn E <-> ( F u. { <. X , Y >. } ) Fn E )
13		fnunop.e	. . . 4 |- E = ( D u. { X } )
14	13	fneq2i 5416	. . 3 |- ( ( F u. { <. X , Y >. } ) Fn E <-> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
15	12, 14	bitri 184	. 2 |- ( G Fn E <-> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
16	10, 15	sylibr 134	1 |- ( ph -> G Fn E )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   u. cun 3195   i^i cin 3196   (/)c0 3491   {csn 3666   <.cop 3669   Fn wfn 5313

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-in1 617  ax-in2 618  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 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  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 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-fun 5320  df-fn 5321

This theorem is referenced by:  tfrlemisucfn  6470  tfr1onlemsucfn  6486