Theorem fndmdif 5805

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 3355	. . . . 5 |- ( F \ G ) C_ F
2		dmss 4975	. . . . 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 5475	. . . . 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 3298	. . 3 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) C_ A )
7		dfss1 3435	. . 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 2824	. . . . 5 |- x e. _V
10	9	eldm 4973	. . . 4 |- ( x e. dom ( F \ G ) <-> E. y x ( F \ G ) y )
11		eqcom 2240	. . . . . . . 8 |- ( ( F ` x ) = ( G ` x ) <-> ( G ` x ) = ( F ` x ) )
12		fnbrfvb 5735	. . . . . . . 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 480	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) = ( G ` x ) <-> x G ( F ` x ) ) )
15	14	necon3abid 2459	. . . . 5 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( F ` x ) =/= ( G ` x ) <-> -. x G ( F ` x ) ) )
16		funfvex 5707	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
17	16	funfni 5478	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
18	17	adantlr 481	. . . . . 6 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( F ` x ) e. _V )
19		breq2 4129	. . . . . . . 8 |- ( y = ( F ` x ) -> ( x G y <-> x G ( F ` x ) ) )
20	19	notbid 677	. . . . . . 7 |- ( y = ( F ` x ) -> ( -. x G y <-> -. x G ( F ` x ) ) )
21	20	ceqsexgv 2955	. . . . . 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 2240	. . . . . . . . . 10 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
24		fnbrfvb 5735	. . . . . . . . . 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 481	. . . . . . . 8 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
27	26	anbi1d 469	. . . . . . 7 |- ( ( ( F Fn A /\ G Fn A ) /\ x e. A ) -> ( ( y = ( F ` x ) /\ -. x G y ) <-> ( x F y /\ -. x G y ) ) )
28		brdif 4179	. . . . . . 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 1878	. . . . 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 3439	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( A i^i dom ( F \ G ) ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
34	8, 33	eqtr3d 2273	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 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   \ cdif 3217   i^i cin 3219   C_ wss 3220   class class class wbr 4125   dom cdm 4769   Fn wfn 5367   ` cfv 5372

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380

This theorem is referenced by:  fndmdifcom  5806