Theorem fnoprabg 6761

Description: Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.)

Assertion

Ref	Expression
fnoprabg	|- ( A. x A. y ( ph -> E! z ps ) -> { <. <. x , y >. , z >. | ( ph /\ ps ) } Fn { <. x , y >. | ph } )

Distinct variable groups:   x, y, z   ph, z

Allowed substitution hints:   ph( x, y)   ps( x, y, z)

Proof of Theorem fnoprabg

Step	Hyp	Ref	Expression
1		eumo 2499	. . . . . 6 |- ( E! z ps -> E* z ps )
2	1	imim2i 16	. . . . 5 |- ( ( ph -> E! z ps ) -> ( ph -> E* z ps ) )
3		moanimv 2531	. . . . 5 |- ( E* z ( ph /\ ps ) <-> ( ph -> E* z ps ) )
4	2, 3	sylibr 224	. . . 4 |- ( ( ph -> E! z ps ) -> E* z ( ph /\ ps ) )
5	4	2alimi 1740	. . 3 |- ( A. x A. y ( ph -> E! z ps ) -> A. x A. y E* z ( ph /\ ps ) )
6		funoprabg 6759	. . 3 |- ( A. x A. y E* z ( ph /\ ps ) -> Fun { <. <. x , y >. , z >. | ( ph /\ ps ) } )
7	5, 6	syl 17	. 2 |- ( A. x A. y ( ph -> E! z ps ) -> Fun { <. <. x , y >. , z >. | ( ph /\ ps ) } )
8		dmoprab 6741	. . 3 |- dom { <. <. x , y >. , z >. | ( ph /\ ps ) } = { <. x , y >. | E. z ( ph /\ ps ) }
9		nfa1 2028	. . . 4 |- F/ x A. x A. y ( ph -> E! z ps )
10		nfa2 2040	. . . 4 |- F/ y A. x A. y ( ph -> E! z ps )
11		simpl 473	. . . . . . . 8 |- ( ( ph /\ ps ) -> ph )
12	11	exlimiv 1858	. . . . . . 7 |- ( E. z ( ph /\ ps ) -> ph )
13		euex 2494	. . . . . . . . . 10 |- ( E! z ps -> E. z ps )
14	13	imim2i 16	. . . . . . . . 9 |- ( ( ph -> E! z ps ) -> ( ph -> E. z ps ) )
15	14	ancld 576	. . . . . . . 8 |- ( ( ph -> E! z ps ) -> ( ph -> ( ph /\ E. z ps ) ) )
16		19.42v 1918	. . . . . . . 8 |- ( E. z ( ph /\ ps ) <-> ( ph /\ E. z ps ) )
17	15, 16	syl6ibr 242	. . . . . . 7 |- ( ( ph -> E! z ps ) -> ( ph -> E. z ( ph /\ ps ) ) )
18	12, 17	impbid2 216	. . . . . 6 |- ( ( ph -> E! z ps ) -> ( E. z ( ph /\ ps ) <-> ph ) )
19	18	sps 2055	. . . . 5 |- ( A. y ( ph -> E! z ps ) -> ( E. z ( ph /\ ps ) <-> ph ) )
20	19	sps 2055	. . . 4 |- ( A. x A. y ( ph -> E! z ps ) -> ( E. z ( ph /\ ps ) <-> ph ) )
21	9, 10, 20	opabbid 4715	. . 3 |- ( A. x A. y ( ph -> E! z ps ) -> { <. x , y >. | E. z ( ph /\ ps ) } = { <. x , y >. | ph } )
22	8, 21	syl5eq 2668	. 2 |- ( A. x A. y ( ph -> E! z ps ) -> dom { <. <. x , y >. , z >. | ( ph /\ ps ) } = { <. x , y >. | ph } )
23		df-fn 5891	. 2 |- ( { <. <. x , y >. , z >. | ( ph /\ ps ) } Fn { <. x , y >. | ph } <-> ( Fun { <. <. x , y >. , z >. | ( ph /\ ps ) } /\ dom { <. <. x , y >. , z >. | ( ph /\ ps ) } = { <. x , y >. | ph } ) )
24	7, 22, 23	sylanbrc 698	1 |- ( A. x A. y ( ph -> E! z ps ) -> { <. <. x , y >. , z >. | ( ph /\ ps ) } Fn { <. x , y >. | ph } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   E!weu 2470   E*wmo 2471   {copab 4712   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-fun 5890  df-fn 5891  df-oprab 6654

This theorem is referenced by:  fnoprab  6763  ovg  6799