Theorem eqfnfvd 5735

Description: Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
eqfnfvd.1	|- ( ph -> F Fn A )
eqfnfvd.2	|- ( ph -> G Fn A )
eqfnfvd.3	|- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) )

Assertion

Ref	Expression
eqfnfvd	|- ( ph -> F = G )

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

Proof of Theorem eqfnfvd

Step	Hyp	Ref	Expression
1		eqfnfvd.3	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( G ` x ) )
2	1	ralrimiva 2603	. 2 |- ( ph -> A. x e. A ( F ` x ) = ( G ` x ) )
3		eqfnfvd.1	. . 3 |- ( ph -> F Fn A )
4		eqfnfvd.2	. . 3 |- ( ph -> G Fn A )
5		eqfnfv 5732	. . 3 |- ( ( F Fn A /\ G Fn A ) -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
6	3, 4, 5	syl2anc 411	. 2 |- ( ph -> ( F = G <-> A. x e. A ( F ` x ) = ( G ` x ) ) )
7	2, 6	mpbird 167	1 |- ( ph -> F = G )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   A.wral 2508   Fn wfn 5313   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by:  foeqcnvco  5914  f1eqcocnv  5915  offeq  6232  tfrlem1  6454  frecrdg  6554  updjudhcoinlf  7247  updjudhcoinrg  7248  nnnninfeq  7295  seq3val  10682  seqvalcd  10683  seq3feq2  10698  seq3feq  10702  seqfeq3  10751  ccatlid  11141  ccatrid  11142  ccatass  11143  ccatswrd  11202  swrdccat2  11203  pfxid  11218  ccatpfx  11233  pfxccat1  11234  swrdswrd  11237  cats1un  11253  swrdccatin1  11257  swrdccatin2  11261  pfxccatin12  11265  seq3shft  11349  efcvgfsum  12178  nninfctlemfo  12561  xpsfeq  13378  upxp  14946  uptx  14948  dvidlemap  15365  dvidrelem  15366  dvidsslem  15367  dvrecap  15387  peano4nninf  16372  nninfsellemeqinf  16382  nninffeq  16386  refeq  16396