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Theorem fvreseq 5298
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 5039 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) Fn 𝐵)
2 fnssres 5039 . . . 4 ((𝐺 Fn 𝐴𝐵𝐴) → (𝐺𝐵) Fn 𝐵)
31, 2anim12i 325 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ (𝐺 Fn 𝐴𝐵𝐴)) → ((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵))
43anandirs 535 . 2 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵))
5 eqfnfv 5292 . . 3 (((𝐹𝐵) Fn 𝐵 ∧ (𝐺𝐵) Fn 𝐵) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥)))
6 fvres 5225 . . . . 5 (𝑥𝐵 → ((𝐹𝐵)‘𝑥) = (𝐹𝑥))
7 fvres 5225 . . . . 5 (𝑥𝐵 → ((𝐺𝐵)‘𝑥) = (𝐺𝑥))
86, 7eqeq12d 2070 . . . 4 (𝑥𝐵 → (((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥) ↔ (𝐹𝑥) = (𝐺𝑥)))
98ralbiia 2355 . . 3 (∀𝑥𝐵 ((𝐹𝐵)‘𝑥) = ((𝐺𝐵)‘𝑥) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
105, 9syl6bb 189 . 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 101  wb 102   = wceq 1259  wcel 1409  wral 2323  wss 2944  cres 4374   Fn wfn 4924  cfv 4929
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 3902  ax-pow 3954  ax-pr 3971
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 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-res 4384  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937
This theorem is referenced by:  tfri3  5983
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