Theorem fndmdif 5601

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 3253	. . . . 5 |- ( F \ G ) C_ F
2		dmss 4810	. . . . 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 5297	. . . . 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 3197	. . 3 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) C_ A )
7		dfss1 3331	. . 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 2733	. . . . 5 |- x e. _V
10	9	eldm 4808	. . . 4 |- ( x e. dom ( F \ G ) <-> E. y x ( F \ G ) y )
11		eqcom 2172	. . . . . . . 8 |- ( ( F ` x ) = ( G ` x ) <-> ( G ` x ) = ( F ` x ) )
12		fnbrfvb 5537	. . . . . . . 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 473	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) = ( G ` x ) <-> x G ( F ` x ) ) )
15	14	necon3abid 2379	. . . . 5 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) =/= ( G ` x ) <-> -. x G ( F ` x ) ) )
16		funfvex 5513	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
17	16	funfni 5298	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
18	17	adantlr 474	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( F ` x ) e. _V )
19		breq2 3993	. . . . . . . 8 |- ( y = ( F ` x ) -> ( x G y <-> x G ( F ` x ) ) )
20	19	notbid 662	. . . . . . 7 |- ( y = ( F ` x ) -> ( -. x G y <-> -. x G ( F ` x ) ) )
21	20	ceqsexgv 2859	. . . . . 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 2172	. . . . . . . . . 10 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
24		fnbrfvb 5537	. . . . . . . . . 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 474	. . . . . . . 8 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
27	26	anbi1d 462	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( y = ( F ` x ) /\ -. x G y ) <-> ( x F y /\ -. x G y ) ) )
28		brdif 4042	. . . . . . 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 1818	. . . . 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 3335	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( A i^i dom ( F \ G ) ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
34	8, 33	eqtr3d 2205	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 1348   E.wex 1485   e. wcel 2141   =/= wne 2340   {crab 2452   _Vcvv 2730   \ cdif 3118   i^i cin 3120   C_ wss 3121   class class class wbr 3989   dom cdm 4611   Fn wfn 5193   ` cfv 5198

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  fndmdifcom  5602