Theorem fgraphxp 37789

Description: Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Assertion

Ref	Expression
fgraphxp	|- ( F : A --> B -> F = { x e. ( A X. B ) | ( F ` ( 1st ` x ) ) = ( 2nd ` x ) } )

Distinct variable groups:   x, F   x, A   x, B

Proof of Theorem fgraphxp

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fgraphopab 37788	. 2 |- ( F : A --> B -> F = { <. a , b >. | ( ( a e. A /\ b e. B ) /\ ( F ` a ) = b ) } )
2		vex 3203	. . . . . . 7 |- a e. _V
3		vex 3203	. . . . . . 7 |- b e. _V
4	2, 3	op1std 7178	. . . . . 6 |- ( x = <. a , b >. -> ( 1st ` x ) = a )
5	4	fveq2d 6195	. . . . 5 |- ( x = <. a , b >. -> ( F ` ( 1st ` x ) ) = ( F ` a ) )
6	2, 3	op2ndd 7179	. . . . 5 |- ( x = <. a , b >. -> ( 2nd ` x ) = b )
7	5, 6	eqeq12d 2637	. . . 4 |- ( x = <. a , b >. -> ( ( F ` ( 1st ` x ) ) = ( 2nd ` x ) <-> ( F ` a ) = b ) )
8	7	rabxp 5154	. . 3 |- { x e. ( A X. B ) | ( F ` ( 1st ` x ) ) = ( 2nd ` x ) } = { <. a , b >. | ( a e. A /\ b e. B /\ ( F ` a ) = b ) }
9		df-3an 1039	. . . 4 |- ( ( a e. A /\ b e. B /\ ( F ` a ) = b ) <-> ( ( a e. A /\ b e. B ) /\ ( F ` a ) = b ) )
10	9	opabbii 4717	. . 3 |- { <. a , b >. | ( a e. A /\ b e. B /\ ( F ` a ) = b ) } = { <. a , b >. | ( ( a e. A /\ b e. B ) /\ ( F ` a ) = b ) }
11	8, 10	eqtri 2644	. 2 |- { x e. ( A X. B ) | ( F ` ( 1st ` x ) ) = ( 2nd ` x ) } = { <. a , b >. | ( ( a e. A /\ b e. B ) /\ ( F ` a ) = b ) }
12	1, 11	syl6eqr 2674	1 |- ( F : A --> B -> F = { x e. ( A X. B ) | ( F ` ( 1st ` x ) ) = ( 2nd ` x ) } )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   <.cop 4183   {copab 4712   X. cxp 5112   -->wf 5884   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-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-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-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  hausgraph  37790