Theorem fnunsn 5464

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 5402	. . . 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 3750	. . . 4 |- ( ( D i^i { X } ) = (/) <-> -. X e. D )
8	6, 7	sylibr 134	. . 3 |- ( ph -> ( D i^i { X } ) = (/) )
9		fnun 5463	. . 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 1273	. 2 |- ( ph -> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
11		fnunop.g	. . . 4 |- G = ( F u. { <. X , Y >. } )
12	11	fneq1i 5449	. . 3 |- ( G Fn E <-> ( F u. { <. X , Y >. } ) Fn E )
13		fnunop.e	. . . 4 |- E = ( D u. { X } )
14	13	fneq2i 5450	. . 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 1398   e. wcel 2203   _Vcvv 2812   u. cun 3208   i^i cin 3209   (/)c0 3507   {csn 3688   <.cop 3691   Fn wfn 5346

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-fun 5353  df-fn 5354

This theorem is referenced by:  tfrlemisucfn  6554  tfr1onlemsucfn  6570