Theorem fnunsn 5006

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 4947	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> { <. X , Y >. } Fn { X } )
5	2, 3, 4	syl2anc 391	. . 3 |- ( ph -> { <. X , Y >. } Fn { X } )
6		fnunop.d	. . . 4 |- ( ph -> -. X e. D )
7		disjsn 3432	. . . 4 |- ( ( D i^i { X } ) = (/) <-> -. X e. D )
8	6, 7	sylibr 137	. . 3 |- ( ph -> ( D i^i { X } ) = (/) )
9		fnun 5005	. . 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 1134	. 2 |- ( ph -> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
11		fnunop.g	. . . 4 |- G = ( F u. { <. X , Y >. } )
12	11	fneq1i 4993	. . 3 |- ( G Fn E <-> ( F u. { <. X , Y >. } ) Fn E )
13		fnunop.e	. . . 4 |- E = ( D u. { X } )
14	13	fneq2i 4994	. . 3 |- ( ( F u. { <. X , Y >. } ) Fn E <-> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
15	12, 14	bitri 173	. 2 |- ( G Fn E <-> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
16	10, 15	sylibr 137	1 |- ( ph -> G Fn E )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1243   e. wcel 1393   _Vcvv 2557   u. cun 2915   i^i cin 2916   (/)c0 3224   {csn 3375   <.cop 3378   Fn wfn 4897

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-fun 4904  df-fn 4905

This theorem is referenced by:  tfrlemisucfn  5938