Theorem eqfnfv2 5701

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 4897	. . . 4 |- ( F = G -> dom F = dom G )
2		fndm 5392	. . . . 5 |- ( F Fn A -> dom F = A )
3		fndm 5392	. . . . 5 |- ( G Fn B -> dom G = B )
4	2, 3	eqeqan12d 2223	. . . 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 5382	. . . . . 6 |- ( A = B -> ( G Fn A <-> G Fn B ) )
8	7	biimparc 299	. . . . 5 |- ( ( G Fn B /\ A = B ) -> G Fn A )
9		eqfnfv 5700	. . . . 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 1373   A.wral 2486   dom cdm 4693   Fn wfn 5285   ` cfv 5290

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  eqfnfv3  5702  eqfunfv  5705  eqfnov  6075  2ffzeq  10298  eqwrd  11071