Theorem eqfnfv2 5594

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 4811	. . . 4 |- ( F = G -> dom F = dom G )
2		fndm 5297	. . . . 5 |- ( F Fn A -> dom F = A )
3		fndm 5297	. . . . 5 |- ( G Fn B -> dom G = B )
4	2, 3	eqeqan12d 2186	. . . 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 5287	. . . . . 6 |- ( A = B -> ( G Fn A <-> G Fn B ) )
8	7	biimparc 297	. . . . 5 |- ( ( G Fn B /\ A = B ) -> G Fn A )
9		eqfnfv 5593	. . . . 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 449	. 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 1348   A.wral 2448   dom cdm 4611   Fn wfn 5193   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  eqfnfv3  5595  eqfunfv  5598  eqfnov  5959  2ffzeq  10097