Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqfnfv2f Structured version   Unicode version

Theorem eqfnfv2f 5980
 Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5976 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
Hypotheses
Ref Expression
eqfnfv2f.1
eqfnfv2f.2
Assertion
Ref Expression
eqfnfv2f
Distinct variable group:   ,
Allowed substitution hints:   ()   ()

Proof of Theorem eqfnfv2f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqfnfv 5976 . 2
2 eqfnfv2f.1 . . . . 5
3 nfcv 2629 . . . . 5
42, 3nffv 5873 . . . 4
5 eqfnfv2f.2 . . . . 5
65, 3nffv 5873 . . . 4
74, 6nfeq 2640 . . 3
8 nfv 1683 . . 3
9 fveq2 5866 . . . 4
10 fveq2 5866 . . . 4
119, 10eqeq12d 2489 . . 3
127, 8, 11cbvral 3084 . 2
131, 12syl6bb 261 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379  wnfc 2615  wral 2814   wfn 5583  cfv 5588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator