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Theorem eqfnfv 5208
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 A 𝐺 Fn A) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))
Distinct variable groups:   x,A   x,𝐹   x,𝐺

Proof of Theorem eqfnfv
StepHypRef Expression
1 dffn5im 5162 . . 3 (𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))
2 dffn5im 5162 . . 3 (𝐺 Fn A𝐺 = (x A ↦ (𝐺x)))
31, 2eqeqan12d 2052 . 2 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺 ↔ (x A ↦ (𝐹x)) = (x A ↦ (𝐺x))))
4 funfvex 5135 . . . . . 6 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
54funfni 4942 . . . . 5 ((𝐹 Fn A x A) → (𝐹x) V)
65ralrimiva 2386 . . . 4 (𝐹 Fn Ax A (𝐹x) V)
7 mpteqb 5204 . . . 4 (x A (𝐹x) V → ((x A ↦ (𝐹x)) = (x A ↦ (𝐺x)) ↔ x A (𝐹x) = (𝐺x)))
86, 7syl 14 . . 3 (𝐹 Fn A → ((x A ↦ (𝐹x)) = (x A ↦ (𝐺x)) ↔ x A (𝐹x) = (𝐺x)))
98adantr 261 . 2 ((𝐹 Fn A 𝐺 Fn A) → ((x A ↦ (𝐹x)) = (x A ↦ (𝐺x)) ↔ x A (𝐹x) = (𝐺x)))
103, 9bitrd 177 1 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺x A (𝐹x) = (𝐺x)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  wral 2300  Vcvv 2551  cmpt 3809   Fn wfn 4840  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  eqfnfv2  5209  eqfnfvd  5211  eqfnfv2f  5212  fvreseq  5214  fneqeql  5218  fconst2g  5319  cocan1  5370  cocan2  5371  tfri3  5894
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