Theorem fndmin 5754

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 5691	. . . . . 6 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2		df-mpt 4152	. . . . . 6 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
3	1, 2	eqtrdi 2280	. . . . 5 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
4		dffn5im 5691	. . . . . 6 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
5		df-mpt 4152	. . . . . 6 |- ( x e. A |-> ( G ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) }
6	4, 5	eqtrdi 2280	. . . . 5 |- ( G Fn A -> G = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) } )
7	3, 6	ineqan12d 3410	. . . 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 4862	. . . 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 2280	. . 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 4933	. 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 594	. . . . . . . 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 1653	. . . . . . 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 1955	. . . . . . 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 186	. . . . . 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 5656	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
16		eqeq1 2238	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( y = ( G ` x ) <-> ( F ` x ) = ( G ` x ) ) )
17	16	ceqsexgv 2935	. . . . . . . . 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 5432	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( E. y ( y = ( F ` x ) /\ y = ( G ` x ) ) <-> ( F ` x ) = ( G ` x ) ) )
20	19	pm5.32da 452	. . . . . 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	bitrid 192	. . . . 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 2349	. . . 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 4942	. . . 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 2519	. . . 4 |- { x e. A | ( F ` x ) = ( G ` x ) } = { x | ( x e. A /\ ( F ` x ) = ( G ` x ) ) }
25	22, 23, 24	3eqtr4g 2289	. . 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 276	. 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 2264	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) = { x e. A | ( F ` x ) = ( G ` x ) } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   {crab 2514   _Vcvv 2802   i^i cin 3199   {copab 4149   |-> cmpt 4150   dom cdm 4725   Fun wfun 5320   Fn wfn 5321   ` cfv 5326

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by:  fneqeql  5755  mhmeql  13574  ghmeql  13853