Theorem fndmin 5592

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

Assertion

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

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

Proof of Theorem fndmin

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5im 5532	. . . . . 6 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2		df-mpt 4045	. . . . . 6 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
3	1, 2	eqtrdi 2215	. . . . 5 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
4		dffn5im 5532	. . . . . 6 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
5		df-mpt 4045	. . . . . 6 |- ( x e. A |-> ( G ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) }
6	4, 5	eqtrdi 2215	. . . . 5 |- ( G Fn A -> G = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) } )
7	3, 6	ineqan12d 3325	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> ( F i^i G ) = ( { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } i^i { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) } ) )
8		inopab 4736	. . . 4 |- ( { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } i^i { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) } ) = { <. x , y >. | ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) }
9	7, 8	eqtrdi 2215	. . 3 |- ( ( F Fn A /\ G Fn A ) -> ( F i^i G ) = { <. x , y >. | ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) } )
10	9	dmeqd 4806	. 2 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) = dom { <. x , y >. | ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) } )
11		anandi 580	. . . . . . . 8 |- ( ( x e. A /\ ( y = ( F ` x ) /\ y = ( G ` x ) ) ) <-> ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) )
12	11	exbii 1593	. . . . . . 7 |- ( E. y ( x e. A /\ ( y = ( F ` x ) /\ y = ( G ` x ) ) ) <-> E. y ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) )
13		19.42v 1894	. . . . . . 7 |- ( E. y ( x e. A /\ ( y = ( F ` x ) /\ y = ( G ` x ) ) ) <-> ( x e. A /\ E. y ( y = ( F ` x ) /\ y = ( G ` x ) ) ) )
14	12, 13	bitr3i 185	. . . . . 6 |- ( E. y ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) <-> ( x e. A /\ E. y ( y = ( F ` x ) /\ y = ( G ` x ) ) ) )
15		funfvex 5503	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
16		eqeq1 2172	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( y = ( G ` x ) <-> ( F ` x ) = ( G ` x ) ) )
17	16	ceqsexgv 2855	. . . . . . . . 9 |- ( ( F ` x ) e. _V -> ( E. y ( y = ( F ` x ) /\ y = ( G ` x ) ) <-> ( F ` x ) = ( G ` x ) ) )
18	15, 17	syl 14	. . . . . . . 8 |- ( ( Fun F /\ x e. dom F ) -> ( E. y ( y = ( F ` x ) /\ y = ( G ` x ) ) <-> ( F ` x ) = ( G ` x ) ) )
19	18	funfni 5288	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( E. y ( y = ( F ` x ) /\ y = ( G ` x ) ) <-> ( F ` x ) = ( G ` x ) ) )
20	19	pm5.32da 448	. . . . . 6 |- ( F Fn A -> ( ( x e. A /\ E. y ( y = ( F ` x ) /\ y = ( G ` x ) ) ) <-> ( x e. A /\ ( F ` x ) = ( G ` x ) ) ) )
21	14, 20	syl5bb 191	. . . . 5 |- ( F Fn A -> ( E. y ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) <-> ( x e. A /\ ( F ` x ) = ( G ` x ) ) ) )
22	21	abbidv 2284	. . . 4 |- ( F Fn A -> { x | E. y ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) } = { x | ( x e. A /\ ( F ` x ) = ( G ` x ) ) } )
23		dmopab 4815	. . . 4 |- dom { <. x , y >. | ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) } = { x | E. y ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) }
24		df-rab 2453	. . . 4 |- { x e. A | ( F ` x ) = ( G ` x ) } = { x | ( x e. A /\ ( F ` x ) = ( G ` x ) ) }
25	22, 23, 24	3eqtr4g 2224	. . 3 |- ( F Fn A -> dom { <. x , y >. | ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) } = { x e. A | ( F ` x ) = ( G ` x ) } )
26	25	adantr 274	. 2 |- ( ( F Fn A /\ G Fn A ) -> dom { <. x , y >. | ( ( x e. A /\ y = ( F ` x ) ) /\ ( x e. A /\ y = ( G ` x ) ) ) } = { x e. A | ( F ` x ) = ( G ` x ) } )
27	10, 26	eqtrd 2198	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) = { x e. A | ( F ` x ) = ( G ` x ) } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   {cab 2151   {crab 2448   _Vcvv 2726   i^i cin 3115   {copab 4042   |-> cmpt 4043   dom cdm 4604   Fun wfun 5182   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-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-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  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-mpt 4045  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:  fneqeql  5593