Theorem eqfnfv2 4767

Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (The proof was shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

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

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

Proof of Theorem eqfnfv2

Step	Hyp	Ref	Expression
1		eqfnfv 4766	. 2 |- ((F Fn A /\ G Fn A) -> (F = G <-> (A = A /\ A.x e. A (F` x) = (G` x))))
2		eqid 1884	. . 3 |- A = A
3	2	biantrur 794	. 2 |- (A.x e. A (F` x) = (G` x) <-> (A = A /\ A.x e. A (F` x) = (G` x)))
4	1, 3	syl6bbr 597	1 |- ((F Fn A /\ G Fn A) -> (F = G <-> A.x e. A (F` x) = (G` x)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298  A.wral 2105   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  eqfnfv2f 4770  fvreseq 4772  fconst2g 4821  curry1 5075  curry2 5078  mapenlem2 5584  seq1res 7740  seq1shftid 7769  seq1seqz 7784  seq1seq0 7788  seqzeq 7798  seqzres 7803  seqzres2 7804  invfval 9593  sspn 9734  nmlno0lem 9793  phoeqi 9859  sinco 10016  cosco 10017  shftefif1olem 10095  dfiop2 11316  hoeq 11323  ho01i 11391  hoeq1 11393  kbpj 11517  nmlnop0iALT 11557  lnopco0i 11566  lnopconi 11600  lnfnconi 11627  hmopidmpji 11724  pjssdif2i 11746  pjinvari 11764  bnj1524 13177  bnj580 13301  cayleylem2 13642  surjsec2 14467  repfuntw 14502  cocanfo 15689  eqfnun 15705  addrcom 16475  stb2xpl 16742  stb3xpl 16743

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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