Theorem fndmdif 5590

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 3248	. . . . 5 |- ( F \ G ) C_ F
2		dmss 4803	. . . . 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 5287	. . . . 5 |- ( F Fn A -> dom F = A )
5	4	adantr 274	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> dom F = A )
6	3, 5	sseqtrid 3192	. . 3 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) C_ A )
7		dfss1 3326	. . 3 |- ( dom ( F \ G ) C_ A <-> ( A i^i dom ( F \ G ) ) = dom ( F \ G ) )
8	6, 7	sylib 121	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( A i^i dom ( F \ G ) ) = dom ( F \ G ) )
9		vex 2729	. . . . 5 |- x e. _V
10	9	eldm 4801	. . . 4 |- ( x e. dom ( F \ G ) <-> E. y x ( F \ G ) y )
11		eqcom 2167	. . . . . . . 8 |- ( ( F ` x ) = ( G ` x ) <-> ( G ` x ) = ( F ` x ) )
12		fnbrfvb 5527	. . . . . . . 8 |- ( ( G Fn A /\ x e. A ) -> ( ( G ` x ) = ( F ` x ) <-> x G ( F ` x ) ) )
13	11, 12	syl5bb 191	. . . . . . 7 |- ( ( G Fn A /\ x e. A ) -> ( ( F ` x ) = ( G ` x ) <-> x G ( F ` x ) ) )
14	13	adantll 468	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) = ( G ` x ) <-> x G ( F ` x ) ) )
15	14	necon3abid 2375	. . . . 5 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) =/= ( G ` x ) <-> -. x G ( F ` x ) ) )
16		funfvex 5503	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
17	16	funfni 5288	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
18	17	adantlr 469	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( F ` x ) e. _V )
19		breq2 3986	. . . . . . . 8 |- ( y = ( F ` x ) -> ( x G y <-> x G ( F ` x ) ) )
20	19	notbid 657	. . . . . . 7 |- ( y = ( F ` x ) -> ( -. x G y <-> -. x G ( F ` x ) ) )
21	20	ceqsexgv 2855	. . . . . 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 2167	. . . . . . . . . 10 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
24		fnbrfvb 5527	. . . . . . . . . 10 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
25	23, 24	syl5bb 191	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
26	25	adantlr 469	. . . . . . . 8 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
27	26	anbi1d 461	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( y = ( F ` x ) /\ -. x G y ) <-> ( x F y /\ -. x G y ) ) )
28		brdif 4035	. . . . . . 7 |- ( x ( F \ G ) y <-> ( x F y /\ -. x G y ) )
29	27, 28	bitr4di 197	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( y = ( F ` x ) /\ -. x G y ) <-> x ( F \ G ) y ) )
30	29	exbidv 1813	. . . . 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 216	. . . 4 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( E. y x ( F \ G ) y <-> ( F ` x ) =/= ( G ` x ) ) )
32	10, 31	syl5bb 191	. . 3 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( x e. dom ( F \ G ) <-> ( F ` x ) =/= ( G ` x ) ) )
33	32	rabbi2dva 3330	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( A i^i dom ( F \ G ) ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
34	8, 33	eqtr3d 2200	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   =/= wne 2336   {crab 2448   _Vcvv 2726   \ cdif 3113   i^i cin 3115   C_ wss 3116   class class class wbr 3982   dom cdm 4604   Fn wfn 5183   ` cfv 5188

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  fndmdifcom  5591