ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  eqfnfv GIF version

Theorem eqfnfv 5293
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 5247 . . 3 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
2 dffn5im 5247 . . 3 (𝐺 Fn 𝐴𝐺 = (𝑥𝐴 ↦ (𝐺𝑥)))
31, 2eqeqan12d 2071 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐺𝑥))))
4 funfvex 5220 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
54funfni 5027 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → (𝐹𝑥) ∈ V)
65ralrimiva 2409 . . . 4 (𝐹 Fn 𝐴 → ∀𝑥𝐴 (𝐹𝑥) ∈ V)
7 mpteqb 5289 . . . 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 265 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝑥𝐴 ↦ (𝐹𝑥)) = (𝑥𝐴 ↦ (𝐺𝑥)) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
103, 9bitrd 181 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  Vcvv 2574  cmpt 3846   Fn wfn 4925  cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by:  eqfnfv2  5294  eqfnfvd  5296  eqfnfv2f  5297  fvreseq  5299  fneqeql  5303  fconst2g  5404  cocan1  5455  cocan2  5456  tfri3  5984  iser0f  9416
  Copyright terms: Public domain W3C validator