Theorem fndmdif 6321

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 3737	. . . . 5 |- ( F \ G ) C_ F
2		dmss 5323	. . . . 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 5990	. . . . 5 |- ( F Fn A -> dom F = A )
5	4	adantr 481	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> dom F = A )
6	3, 5	syl5sseq 3653	. . 3 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) C_ A )
7		sseqin2 3817	. . 3 |- ( dom ( F \ G ) C_ A <-> ( A i^i dom ( F \ G ) ) = dom ( F \ G ) )
8	6, 7	sylib 208	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( A i^i dom ( F \ G ) ) = dom ( F \ G ) )
9		vex 3203	. . . . 5 |- x e. _V
10	9	eldm 5321	. . . 4 |- ( x e. dom ( F \ G ) <-> E. y x ( F \ G ) y )
11		eqcom 2629	. . . . . . . . 9 |- ( ( F ` x ) = ( G ` x ) <-> ( G ` x ) = ( F ` x ) )
12		fnbrfvb 6236	. . . . . . . . 9 |- ( ( G Fn A /\ x e. A ) -> ( ( G ` x ) = ( F ` x ) <-> x G ( F ` x ) ) )
13	11, 12	syl5bb 272	. . . . . . . 8 |- ( ( G Fn A /\ x e. A ) -> ( ( F ` x ) = ( G ` x ) <-> x G ( F ` x ) ) )
14	13	adantll 750	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) = ( G ` x ) <-> x G ( F ` x ) ) )
15	14	necon3abid 2830	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) =/= ( G ` x ) <-> -. x G ( F ` x ) ) )
16		fvex 6201	. . . . . . 7 |- ( F ` x ) e. _V
17		breq2 4657	. . . . . . . 8 |- ( y = ( F ` x ) -> ( x G y <-> x G ( F ` x ) ) )
18	17	notbid 308	. . . . . . 7 |- ( y = ( F ` x ) -> ( -. x G y <-> -. x G ( F ` x ) ) )
19	16, 18	ceqsexv 3242	. . . . . 6 |- ( E. y ( y = ( F ` x ) /\ -. x G y ) <-> -. x G ( F ` x ) )
20	15, 19	syl6bbr 278	. . . . 5 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) =/= ( G ` x ) <-> E. y ( y = ( F ` x ) /\ -. x G y ) ) )
21		eqcom 2629	. . . . . . . . . 10 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
22		fnbrfvb 6236	. . . . . . . . . 10 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
23	21, 22	syl5bb 272	. . . . . . . . 9 |- ( ( F Fn A /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
24	23	adantlr 751	. . . . . . . 8 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
25	24	anbi1d 741	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( y = ( F ` x ) /\ -. x G y ) <-> ( x F y /\ -. x G y ) ) )
26		brdif 4705	. . . . . . 7 |- ( x ( F \ G ) y <-> ( x F y /\ -. x G y ) )
27	25, 26	syl6bbr 278	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( y = ( F ` x ) /\ -. x G y ) <-> x ( F \ G ) y ) )
28	27	exbidv 1850	. . . . 5 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( E. y ( y = ( F ` x ) /\ -. x G y ) <-> E. y x ( F \ G ) y ) )
29	20, 28	bitr2d 269	. . . 4 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( E. y x ( F \ G ) y <-> ( F ` x ) =/= ( G ` x ) ) )
30	10, 29	syl5bb 272	. . 3 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( x e. dom ( F \ G ) <-> ( F ` x ) =/= ( G ` x ) ) )
31	30	rabbi2dva 3821	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( A i^i dom ( F \ G ) ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
32	8, 31	eqtr3d 2658	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   class class class wbr 4653   dom cdm 5114   Fn wfn 5883   ` cfv 5888

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-ne 2795  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-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  fndmdifcom  6322  fndmdifeq0  6323  fndifnfp  6442  wemapsolem  8455  wemapso2lem  8457  dsmmbas2  20081  frlmbas  20099  ptcmplem2  21857