Theorem fndmin 5790

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 5727	. . . . . 6 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
2		df-mpt 4178	. . . . . 6 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
3	1, 2	eqtrdi 2283	. . . . 5 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
4		dffn5im 5727	. . . . . 6 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
5		df-mpt 4178	. . . . . 6 |- ( x e. A |-> ( G ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) }
6	4, 5	eqtrdi 2283	. . . . 5 |- ( G Fn A -> G = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) } )
7	3, 6	ineqan12d 3428	. . . 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 4892	. . . 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 2283	. . 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 4963	. 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 1654	. . . . . . 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 1958	. . . . . . 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 5692	. . . . . . . . 9 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
16		eqeq1 2241	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( y = ( G ` x ) <-> ( F ` x ) = ( G ` x ) ) )
17	16	ceqsexgv 2949	. . . . . . . . 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 5463	. . . . . . 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 2354	. . . 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 4972	. . . 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 2531	. . . 4 |- { x e. A | ( F ` x ) = ( G ` x ) } = { x | ( x e. A /\ ( F ` x ) = ( G ` x ) ) }
25	22, 23, 24	3eqtr4g 2292	. . 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 2267	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 1398   E.wex 1541   e. wcel 2205   {cab 2220   {crab 2526   _Vcvv 2815   i^i cin 3213   {copab 4175   |-> cmpt 4176   dom cdm 4754   Fun wfun 5351   Fn wfn 5352   ` cfv 5357

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365

This theorem is referenced by:  fneqeql  5791  mhmeql  13747  ghmeql  14020