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Theorem eqfnfv2 5266
 Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
eqfnfv2
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem eqfnfv2
StepHypRef Expression
1 dmeq 4535 . . . 4
2 fndm 4998 . . . . 5
3 fndm 4998 . . . . 5
42, 3eqeqan12d 2055 . . . 4
51, 4syl5ib 143 . . 3
65pm4.71rd 374 . 2
7 fneq2 4988 . . . . . 6
87biimparc 283 . . . . 5
9 eqfnfv 5265 . . . . 5
108, 9sylan2 270 . . . 4
1110anassrs 380 . . 3
1211pm5.32da 425 . 2
136, 12bitrd 177 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wral 2306   cdm 4345   wfn 4897  cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910 This theorem is referenced by:  eqfnfv3  5267  eqfunfv  5270  eqfnov  5607  2ffzeq  8998
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