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Theorem fnovim 5551
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 = (x A, y B ↦ (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 5162 . 2 (𝐹 Fn (A × B) → 𝐹 = (z (A × B) ↦ (𝐹z)))
2 fveq2 5121 . . . . 5 (z = ⟨x, y⟩ → (𝐹z) = (𝐹‘⟨x, y⟩))
3 df-ov 5458 . . . . 5 (x𝐹y) = (𝐹‘⟨x, y⟩)
42, 3syl6eqr 2087 . . . 4 (z = ⟨x, y⟩ → (𝐹z) = (x𝐹y))
54mpt2mpt 5538 . . 3 (z (A × B) ↦ (𝐹z)) = (x A, y B ↦ (x𝐹y))
65eqeq2i 2047 . 2 (𝐹 = (z (A × B) ↦ (𝐹z)) ↔ 𝐹 = (x A, y B ↦ (x𝐹y)))
71, 6sylib 127 1 (𝐹 Fn (A × B) → 𝐹 = (x A, y B ↦ (x𝐹y)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cop 3370  cmpt 3809   × cxp 4286   Fn wfn 4840  cfv 4845  (class class class)co 5455  cmpt2 5457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460
This theorem is referenced by:  dfioo2  8613
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