Theorem fndmin 5306

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 5251	. . . . . 6 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2		df-mpt 3849	. . . . . 6 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
3	1, 2	syl6eq 2130	. . . . 5 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
4		dffn5im 5251	. . . . . 6 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
5		df-mpt 3849	. . . . . 6 |- ( x e. A |-> ( G ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) }
6	4, 5	syl6eq 2130	. . . . 5 |- ( G Fn A -> G = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) } )
7	3, 6	ineqan12d 3176	. . . 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 4496	. . . 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	syl6eq 2130	. . 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 4565	. 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 555	. . . . . . . 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 1537	. . . . . . 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 1828	. . . . . . 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 184	. . . . . 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 5223	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
16		eqeq1 2088	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( y = ( G ` x ) <-> ( F ` x ) = ( G ` x ) ) )
17	16	ceqsexgv 2725	. . . . . . . . 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 5030	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( E. y ( y = ( F ` x ) /\ y = ( G ` x ) ) <-> ( F ` x ) = ( G ` x ) ) )
20	19	pm5.32da 440	. . . . . 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 190	. . . . 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 2197	. . . 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 4574	. . . 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 2358	. . . 4 |- { x e. A | ( F ` x ) = ( G ` x ) } = { x | ( x e. A /\ ( F ` x ) = ( G ` x ) ) }
25	22, 23, 24	3eqtr4g 2139	. . 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 270	. 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 2114	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F i^i G ) = { x e. A | ( F ` x ) = ( G ` x ) } )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1285   E.wex 1422   e. wcel 1434   {cab 2068   {crab 2353   _Vcvv 2602   i^i cin 2973   {copab 3846   |-> cmpt 3847   dom cdm 4371   Fun wfun 4926   Fn wfn 4927   ` cfv 4932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940

This theorem is referenced by:  fneqeql  5307