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Theorem fvreseq 6478
 Description: Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.) (Proof shortened by AV, 4-Mar-2019.)
Assertion
Ref Expression
fvreseq (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐺
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fvreseq
StepHypRef Expression
1 fvreseq0 6476 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐵𝐴𝐵𝐴)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
21anabsan2 898 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1628  ∀wral 3046   ⊆ wss 3711   ↾ cres 5264   Fn wfn 6040  ‘cfv 6045 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-fv 6053 This theorem is referenced by:  wfr3g  7578  tfr3  7660  fseqenlem1  9033  symgfixf1  18053  dchrresb  25179  bnj1536  31227  bnj1253  31388  bnj1280  31391  rdgprc  32001  frr3g  32081  fourierdlem73  40895
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