Theorem foov 5910

Description: An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.)

Assertion

Ref	Expression
foov	|- ( F : ( A X. B ) -onto-> C <-> ( F : ( A X. B ) --> C /\ A. z e. C E. x e. A E. y e. B z = ( x F y ) ) )

Distinct variable groups:   x, y, z, A   x, B, y, z   z, C   x, F, y, z

Allowed substitution hints:   C( x, y)

Proof of Theorem foov

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffo3 5560	. 2 |- ( F : ( A X. B ) -onto-> C <-> ( F : ( A X. B ) --> C /\ A. z e. C E. w e. ( A X. B ) z = ( F ` w ) ) )
2		fveq2 5414	. . . . . . 7 |- ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3		df-ov 5770	. . . . . . 7 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	syl6eqr 2188	. . . . . 6 |- ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5	4	eqeq2d 2149	. . . . 5 |- ( w = <. x , y >. -> ( z = ( F ` w ) <-> z = ( x F y ) ) )
6	5	rexxp 4678	. . . 4 |- ( E. w e. ( A X. B ) z = ( F ` w ) <-> E. x e. A E. y e. B z = ( x F y ) )
7	6	ralbii 2439	. . 3 |- ( A. z e. C E. w e. ( A X. B ) z = ( F ` w ) <-> A. z e. C E. x e. A E. y e. B z = ( x F y ) )
8	7	anbi2i 452	. 2 |- ( ( F : ( A X. B ) --> C /\ A. z e. C E. w e. ( A X. B ) z = ( F ` w ) ) <-> ( F : ( A X. B ) --> C /\ A. z e. C E. x e. A E. y e. B z = ( x F y ) ) )
9	1, 8	bitri 183	1 |- ( F : ( A X. B ) -onto-> C <-> ( F : ( A X. B ) --> C /\ A. z e. C E. x e. A E. y e. B z = ( x F y ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   A.wral 2414   E.wrex 2415   <.cop 3525   X. cxp 4532   -->wf 5114   -onto->wfo 5116   ` cfv 5118  (class class class)co 5767

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-ov 5770

This theorem is referenced by: (None)