Theorem oprabexd 6799

Description: Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
oprabexd.1	|- ( ph -> A e. _V )
oprabexd.2	|- ( ph -> B e. _V )
oprabexd.3	|- ( ( ph /\ ( x e. A /\ y e. B ) ) -> E* z ps )
oprabexd.4	|- ( ph -> F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } )

Assertion

Ref	Expression
oprabexd	|- ( ph -> F e. _V )

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

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

Proof of Theorem oprabexd

Step	Hyp	Ref	Expression
1		oprabexd.4	. 2 |- ( ph -> F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } )
2		oprabexd.3	. . . . . . 7 |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> E* z ps )
3	2	ex 441	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. B ) -> E* z ps ) )
4		moanimv 2380	. . . . . 6 |- ( E* z ( ( x e. A /\ y e. B ) /\ ps ) <-> ( ( x e. A /\ y e. B ) -> E* z ps ) )
5	3, 4	sylibr 217	. . . . 5 |- ( ph -> E* z ( ( x e. A /\ y e. B ) /\ ps ) )
6	5	alrimivv 1782	. . . 4 |- ( ph -> A. x A. y E* z ( ( x e. A /\ y e. B ) /\ ps ) )
7		funoprabg 6414	. . . 4 |- ( A. x A. y E* z ( ( x e. A /\ y e. B ) /\ ps ) -> Fun { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } )
8	6, 7	syl 17	. . 3 |- ( ph -> Fun { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } )
9		dmoprabss 6397	. . . 4 |- dom { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } C_ ( A X. B )
10		oprabexd.1	. . . . 5 |- ( ph -> A e. _V )
11		oprabexd.2	. . . . 5 |- ( ph -> B e. _V )
12		xpexg 6612	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> ( A X. B ) e. _V )
13	10, 11, 12	syl2anc 673	. . . 4 |- ( ph -> ( A X. B ) e. _V )
14		ssexg 4542	. . . 4 |- ( ( dom { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } C_ ( A X. B ) /\ ( A X. B ) e. _V ) -> dom { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } e. _V )
15	9, 13, 14	sylancr 676	. . 3 |- ( ph -> dom { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } e. _V )
16		funex 6149	. . 3 |- ( ( Fun { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } /\ dom { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } e. _V ) -> { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } e. _V )
17	8, 15, 16	syl2anc 673	. 2 |- ( ph -> { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } e. _V )
18	1, 17	eqeltrd 2549	1 |- ( ph -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 376   A.wal 1450   = wceq 1452   e. wcel 1904   E*wmo 2320   _Vcvv 3031   C_ wss 3390   X. cxp 4837   dom cdm 4839   Fun wfun 5583   {coprab 6309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-oprab 6312

This theorem is referenced by: (None)