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Theorem eqfunfv 5323
Description: Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
Assertion
Ref Expression
eqfunfv ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = (𝐺𝑥))))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐺

Proof of Theorem eqfunfv
StepHypRef Expression
1 funfn 4982 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 funfn 4982 . 2 (Fun 𝐺𝐺 Fn dom 𝐺)
3 eqfnfv2 5319 . 2 ((𝐹 Fn dom 𝐹𝐺 Fn dom 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = (𝐺𝑥))))
41, 2, 3syl2anb 285 1 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = (𝐺𝑥))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wral 2353  dom cdm 4392  Fun wfun 4947   Fn wfn 4948  cfv 4953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2826  df-csb 2919  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fn 4956  df-fv 4961
This theorem is referenced by: (None)
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