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Theorem fvreseq 5532
Description: Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.)
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 fnssres 5244 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fnssres 5244 . . . 4 ((𝐺 Fn 𝐴𝐵𝐴) → (𝐺𝐵) Fn 𝐵)
31, 2anim12i 336 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ (𝐺 Fn 𝐴𝐵𝐴)) → ((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵))
43anandirs 583 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵))
5 eqfnfv 5526 . . 3 (((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥)))
6 fvres 5453 . . . . 5 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
7 fvres 5453 . . . . 5 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
86, 7eqeq12d 2155 . . . 4 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥) ↔ (𝐹𝑥) = (𝐺𝑥)))
98ralbiia 2452 . . 3 (∀𝑥𝐵 ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
105, 9syl6bb 195 . 2 (((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
114, 10syl 14 1 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  wral 2417  wss 3076  cres 4549   Fn wfn 5126  cfv 5131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139
This theorem is referenced by:  tfri3  6272
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