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Theorem fnovim 5688
Description: Representation of a function in terms of its values. (Contributed by Jim Kingdon, 16-Jan-2019.)
Assertion
Ref Expression
fnovim (𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦

Proof of Theorem fnovim
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dffn5im 5295 . 2 (𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)))
2 fveq2 5253 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 5594 . . . . 5 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2133 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝑥𝐹𝑦))
54mpt2mpt 5675 . . 3 (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦))
65eqeq2i 2093 . 2 (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
71, 6sylib 120 1 (𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  cop 3425  cmpt 3865   × cxp 4399   Fn wfn 4964  cfv 4969  (class class class)co 5591  cmpt2 5593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-csb 2920  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-iota 4934  df-fun 4971  df-fn 4972  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596
This theorem is referenced by:  mapxpen  6494  dfioo2  9287
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