Theorem eqfnfv2 5578

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 4798	. . . 4 |- ( F = G -> dom F = dom G )
2		fndm 5281	. . . . 5 |- ( F Fn A -> dom F = A )
3		fndm 5281	. . . . 5 |- ( G Fn B -> dom G = B )
4	2, 3	eqeqan12d 2180	. . . 4 |- ( ( F Fn A /\ G Fn B ) -> ( dom F = dom G <-> A = B ) )
5	1, 4	syl5ib 153	. . 3 |- ( ( F Fn A /\ G Fn B ) -> ( F = G -> A = B ) )
6	5	pm4.71rd 392	. 2 |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ F = G ) ) )
7		fneq2 5271	. . . . . 6 |- ( A = B -> ( G Fn A <-> G Fn B ) )
8	7	biimparc 297	. . . . 5 |- ( ( G Fn B /\ A = B ) -> G Fn A )
9		eqfnfv 5577	. . . . 5 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
10	8, 9	sylan2 284	. . . 4 |- ( ( F Fn A /\ ( G Fn B /\ A = B ) ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
11	10	anassrs 398	. . 3 |- ( ( ( F Fn A /\ G Fn B ) /\ A = B ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
12	11	pm5.32da 448	. 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 187	1 |- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   A.wral 2442   dom cdm 4598   Fn wfn 5177   ` cfv 5182

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fn 5185  df-fv 5190

This theorem is referenced by:  eqfnfv3  5579  eqfunfv  5582  eqfnov  5939  2ffzeq  10066