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Theorem eqfnfv 5190
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 5144 . . 3 (𝐹 Fn A𝐹 = (x A ↦ (𝐹x)))
2 dffn5im 5144 . . 3 (𝐺 Fn A𝐺 = (x A ↦ (𝐺x)))
31, 2eqeqan12d 2037 . 2 ((𝐹 Fn A 𝐺 Fn A) → (𝐹 = 𝐺 ↔ (x A ↦ (𝐹x)) = (x A ↦ (𝐺x))))
4 funfvex 5117 . . . . . 6 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
54funfni 4925 . . . . 5 ((𝐹 Fn A x A) → (𝐹x) V)
65ralrimiva 2370 . . . 4 (𝐹 Fn Ax A (𝐹x) V)
7 mpteqb 5186 . . . 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 1228   wcel 1374  wral 2284  Vcvv 2535  cmpt 3792   Fn wfn 4824  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by:  eqfnfv2  5191  eqfnfvd  5193  eqfnfv2f  5194  fvreseq  5196  fneqeql  5200  fconst2g  5301  cocan1  5352  cocan2  5353  tfri3  5875
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