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Theorem eqfnov 6176
 Description: Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)
Assertion
Ref Expression
eqfnov
Distinct variable groups:   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem eqfnov
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqfnfv2 5828 . 2
2 fveq2 5728 . . . . . 6
3 fveq2 5728 . . . . . 6
42, 3eqeq12d 2450 . . . . 5
5 df-ov 6084 . . . . . 6
6 df-ov 6084 . . . . . 6
75, 6eqeq12i 2449 . . . . 5
84, 7syl6bbr 255 . . . 4
98ralxp 5016 . . 3
109anbi2i 676 . 2
111, 10syl6bb 253 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652  wral 2705  cop 3817   cxp 4876   wfn 5449  cfv 5454  (class class class)co 6081 This theorem is referenced by:  eqfnov2  6177  ssceq  14026  sspg  22227  ssps  22229  sspmlem  22231 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-ov 6084
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