Theorem eqfnfv2 5733

Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
eqfnfv2	|- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )

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

Allowed substitution hint:   B( x)

Proof of Theorem eqfnfv2

Step	Hyp	Ref	Expression
1		dmeq 4923	. . . 4 |- ( F = G -> dom F = dom G )
2		fndm 5420	. . . . 5 |- ( F Fn A -> dom F = A )
3		fndm 5420	. . . . 5 |- ( G Fn B -> dom G = B )
4	2, 3	eqeqan12d 2245	. . . 4 |- ( ( F Fn A /\ G Fn B ) -> ( dom F = dom G <-> A = B ) )
5	1, 4	imbitrid 154	. . 3 |- ( ( F Fn A /\ G Fn B ) -> ( F = G -> A = B ) )
6	5	pm4.71rd 394	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ F = G ) ) )
7		fneq2 5410	. . . . . 6 |- ( A = B -> ( G Fn A <-> G Fn B ) )
8	7	biimparc 299	. . . . 5 |- ( ( G Fn B /\ A = B ) -> G Fn A )
9		eqfnfv 5732	. . . . 5 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
10	8, 9	sylan2 286	. . . 4 |- ( ( F Fn A /\ ( G Fn B /\ A = B ) ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
11	10	anassrs 400	. . 3 |- ( ( ( F Fn A /\ G Fn B ) /\ A = B ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
12	11	pm5.32da 452	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( ( A = B /\ F = G ) <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
13	6, 12	bitrd 188	1 |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   A.wral 2508   dom cdm 4719   Fn wfn 5313   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by:  eqfnfv3  5734  eqfunfv  5737  eqfnov  6111  2ffzeq  10337  eqwrd  11112