Theorem fnunsn 5402

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 5340	. . . 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 3705	. . . 4 |- ( ( D i^i { X } ) = (/) <-> -. X e. D )
8	6, 7	sylibr 134	. . 3 |- ( ph -> ( D i^i { X } ) = (/) )
9		fnun 5401	. . 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 1249	. 2 |- ( ph -> ( F u. { <. X , Y >. } ) Fn ( D u. { X } ) )
11		fnunop.g	. . . 4 |- G = ( F u. { <. X , Y >. } )
12	11	fneq1i 5387	. . 3 |- ( G Fn E <-> ( F u. { <. X , Y >. } ) Fn E )
13		fnunop.e	. . . 4 |- E = ( D u. { X } )
14	13	fneq2i 5388	. . 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 1373   e. wcel 2178   _Vcvv 2776   u. cun 3172   i^i cin 3173   (/)c0 3468   {csn 3643   <.cop 3646   Fn wfn 5285

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-fun 5292  df-fn 5293

This theorem is referenced by:  tfrlemisucfn  6433  tfr1onlemsucfn  6449