Theorem foov 6808

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 6374	. 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 6191	. . . . . . 7 |- ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3		df-ov 6653	. . . . . . 7 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	syl6eqr 2674	. . . . . 6 |- ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5	4	eqeq2d 2632	. . . . 5 |- ( w = <. x , y >. -> ( z = ( F ` w ) <-> z = ( x F y ) ) )
6	5	rexxp 5264	. . . 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 2980	. . 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 730	. 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 264	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:   <-> wb 196   /\ wa 384   = wceq 1483   A.wral 2912   E.wrex 2913   <.cop 4183   X. cxp 5112   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650

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-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-ov 6653

This theorem is referenced by:  iunfictbso  8937  xpsff1o  16228  mndpfo  17314  gafo  17729  isgrpo  27351  isgrpoi  27352  opidonOLD  33651  rngmgmbs4  33730  isgrpda  33754