Theorem fndmdif 5761

Description: Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
fndmdif	|- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )

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

Proof of Theorem fndmdif

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difss 3335	. . . . 5 |- ( F \ G ) C_ F
2		dmss 4936	. . . . 5 |- ( ( F \ G ) C_ F -> dom ( F \ G ) C_ dom F )
3	1, 2	ax-mp 5	. . . 4 |- dom ( F \ G ) C_ dom F
4		fndm 5436	. . . . 5 |- ( F Fn A -> dom F = A )
5	4	adantr 276	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> dom F = A )
6	3, 5	sseqtrid 3278	. . 3 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) C_ A )
7		dfss1 3413	. . 3 |- ( dom ( F \ G ) C_ A <-> ( A i^i dom ( F \ G ) ) = dom ( F \ G ) )
8	6, 7	sylib 122	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( A i^i dom ( F \ G ) ) = dom ( F \ G ) )
9		vex 2806	. . . . 5 |- x e. _V
10	9	eldm 4934	. . . 4 |- ( x e. dom ( F \ G ) <-> E. y x ( F \ G ) y )
11		eqcom 2233	. . . . . . . 8 |- ( ( F ` x ) = ( G ` x ) <-> ( G ` x ) = ( F ` x ) )
12		fnbrfvb 5693	. . . . . . . 8 |- ( ( G Fn A /\ x e. A ) -> ( ( G ` x ) = ( F ` x ) <-> x G ( F ` x ) ) )
13	11, 12	bitrid 192	. . . . . . 7 |- ( ( G Fn A /\ x e. A ) -> ( ( F ` x ) = ( G ` x ) <-> x G ( F ` x ) ) )
14	13	adantll 476	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) = ( G ` x ) <-> x G ( F ` x ) ) )
15	14	necon3abid 2442	. . . . 5 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) =/= ( G ` x ) <-> -. x G ( F ` x ) ) )
16		funfvex 5665	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
17	16	funfni 5439	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
18	17	adantlr 477	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( F ` x ) e. _V )
19		breq2 4097	. . . . . . . 8 |- ( y = ( F ` x ) -> ( x G y <-> x G ( F ` x ) ) )
20	19	notbid 673	. . . . . . 7 |- ( y = ( F ` x ) -> ( -. x G y <-> -. x G ( F ` x ) ) )
21	20	ceqsexgv 2936	. . . . . 6 |- ( ( F ` x ) e. _V -> ( E. y ( y = ( F ` x ) /\ -. x G y ) <-> -. x G ( F ` x ) ) )
22	18, 21	syl 14	. . . . 5 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( E. y ( y = ( F ` x ) /\ -. x G y ) <-> -. x G ( F ` x ) ) )
23		eqcom 2233	. . . . . . . . . 10 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
24		fnbrfvb 5693	. . . . . . . . . 10 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
25	23, 24	bitrid 192	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
26	25	adantlr 477	. . . . . . . 8 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
27	26	anbi1d 465	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( y = ( F ` x ) /\ -. x G y ) <-> ( x F y /\ -. x G y ) ) )
28		brdif 4147	. . . . . . 7 |- ( x ( F \ G ) y <-> ( x F y /\ -. x G y ) )
29	27, 28	bitr4di 198	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( y = ( F ` x ) /\ -. x G y ) <-> x ( F \ G ) y ) )
30	29	exbidv 1873	. . . . 5 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( E. y ( y = ( F ` x ) /\ -. x G y ) <-> E. y x ( F \ G ) y ) )
31	15, 22, 30	3bitr2rd 217	. . . 4 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( E. y x ( F \ G ) y <-> ( F ` x ) =/= ( G ` x ) ) )
32	10, 31	bitrid 192	. . 3 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( x e. dom ( F \ G ) <-> ( F ` x ) =/= ( G ` x ) ) )
33	32	rabbi2dva 3417	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( A i^i dom ( F \ G ) ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
34	8, 33	eqtr3d 2266	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2202   =/= wne 2403   {crab 2515   _Vcvv 2803   \ cdif 3198   i^i cin 3200   C_ wss 3201   class class class wbr 4093   dom cdm 4731   Fn wfn 5328   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  fndmdifcom  5762