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Theorem eqfnfv 5252
Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
eqfnfv ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺

Proof of Theorem eqfnfv
StepHypRef Expression
1 dffn5im 5206 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
2 dffn5im 5206 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
31, 2eqeqan12d 2055 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐺𝑥))))
4 funfvex 5179 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
54funfni 4986 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
65ralrimiva 2389 . . . 4 (𝐹 Fn 𝐴 → ∀𝑥𝐴 (𝐹𝑥) ∈ V)
7 mpteqb 5248 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ V → ((𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐺𝑥)) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
86, 7syl 14 . . 3 (𝐹 Fn 𝐴 → ((𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐺𝑥)) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
98adantr 261 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐺𝑥)) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
103, 9bitrd 177 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  wral 2303  Vcvv 2554  cmpt 3815   Fn wfn 4884  cfv 4889
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 3872  ax-pow 3924  ax-pr 3941
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 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-csb 2850  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-mpt 3817  df-id 4027  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-iota 4854  df-fun 4891  df-fn 4892  df-fv 4897
This theorem is referenced by:  eqfnfv2  5253  eqfnfvd  5255  eqfnfv2f  5256  fvreseq  5258  fneqeql  5262  fconst2g  5363  cocan1  5414  cocan2  5415  tfri3  5940  iser0f  9129
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