Theorem eqfnfv2 5680

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 4879	. . . 4 |- ( F = G -> dom F = dom G )
2		fndm 5374	. . . . 5 |- ( F Fn A -> dom F = A )
3		fndm 5374	. . . . 5 |- ( G Fn B -> dom G = B )
4	2, 3	eqeqan12d 2221	. . . 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 5364	. . . . . 6 |- ( A = B -> ( G Fn A <-> G Fn B ) )
8	7	biimparc 299	. . . . 5 |- ( ( G Fn B /\ A = B ) -> G Fn A )
9		eqfnfv 5679	. . . . 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 2484   dom cdm 4676   Fn wfn 5267   ` cfv 5272

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280

This theorem is referenced by:  eqfnfv3  5681  eqfunfv  5684  eqfnov  6054  2ffzeq  10265  eqwrd  11036