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Theorem eqfnfv2 5557
 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 4867 . . . 4
2 fndm 5281 . . . . 5
3 fndm 5281 . . . . 5
42, 3eqeqan12d 2273 . . . 4
51, 4syl5ib 212 . . 3
65pm4.71rd 619 . 2
7 fneq2 5272 . . . . . 6
87biimparc 475 . . . . 5
9 eqfnfv 5556 . . . . 5
108, 9sylan2 462 . . . 4
1110anassrs 632 . . 3
1211pm5.32da 625 . 2
136, 12bitrd 246 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wa 360   wceq 1619  wral 2518   cdm 4661   wfn 4668  cfv 4673 This theorem is referenced by:  eqfnfv3  5558  eqfunfv  5561  eqfnov  5884  soseq  23623  wfr3g  23624  frr3g  23649  axdenselem4  23707  eqfnung2  24485  sdclem2  25819 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-fv 4689
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