Theorem eqfnfvd 5659

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 2567	. 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 5656	. . 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 1364   e. wcel 2164   A.wral 2472   Fn wfn 5250   ` cfv 5255

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by:  foeqcnvco  5834  f1eqcocnv  5835  offeq  6146  tfrlem1  6363  frecrdg  6463  updjudhcoinlf  7141  updjudhcoinrg  7142  nnnninfeq  7189  seq3val  10534  seqvalcd  10535  seq3feq2  10550  seq3feq  10554  seqfeq3  10603  seq3shft  10985  efcvgfsum  11813  nninfctlemfo  12180  xpsfeq  12931  upxp  14451  uptx  14453  dvidlemap  14870  dvidrelem  14871  dvidsslem  14872  dvrecap  14892  peano4nninf  15566  nninfsellemeqinf  15576  nninffeq  15580  refeq  15588