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Theorem fnovim 5811
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 5399 . 2 (𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)))
2 fveq2 5353 . . . . 5 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝐹‘⟨𝑥, 𝑦⟩))
3 df-ov 5709 . . . . 5 (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
42, 3syl6eqr 2150 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (𝐹𝑧) = (𝑥𝐹𝑦))
54mpompt 5795 . . 3 (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦))
65eqeq2i 2110 . 2 (𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
71, 6sylib 121 1 (𝐹 Fn (𝐴 × 𝐵) → 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑥𝐹𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  cop 3477  cmpt 3929   × cxp 4475   Fn wfn 5054  cfv 5059  (class class class)co 5706  cmpo 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711
This theorem is referenced by:  mapxpen  6671  dfioo2  9598  cnmpt22f  12245  cnmptcom  12248  bdxmet  12429
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