Theorem fnunsn 5305

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 5245	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> { <. X , Y >. } Fn { X } )
5	2, 3, 4	syl2anc 409	. . 3 |- ( ph -> { <. X , Y >. } Fn { X } )
6		fnunop.d	. . . 4 |- ( ph -> -. X e. D )
7		disjsn 3645	. . . 4 |- ( ( D i^i { X } ) = (/) <-> -. X e. D )
8	6, 7	sylibr 133	. . 3 |- ( ph -> ( D i^i { X } ) = (/) )
9		fnun 5304	. . 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 1232	. 2 |- ( ph -> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
11		fnunop.g	. . . 4 |- G = ( F u. { <. X , Y >. } )
12	11	fneq1i 5292	. . 3 |- ( G Fn E <-> ( F u. { <. X , Y >. } ) Fn E )
13		fnunop.e	. . . 4 |- E = ( D u. { X } )
14	13	fneq2i 5293	. . 3 |- ( ( F u. { <. X , Y >. } ) Fn E <-> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
15	12, 14	bitri 183	. 2 |- ( G Fn E <-> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
16	10, 15	sylibr 133	1 |- ( ph -> G Fn E )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1348   e. wcel 2141   _Vcvv 2730   u. cun 3119   i^i cin 3120   (/)c0 3414   {csn 3583   <.cop 3586   Fn wfn 5193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-fun 5200  df-fn 5201

This theorem is referenced by:  tfrlemisucfn  6303  tfr1onlemsucfn  6319