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Theorem fovrn 5674
Description: An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.)
Assertion
Ref Expression
fovrn ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

Proof of Theorem fovrn
StepHypRef Expression
1 opelxpi 4402 . . 3 ((𝐴𝑅𝐵𝑆) → ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆))
2 df-ov 5546 . . . 4 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
3 ffvelrn 5332 . . . 4 ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)) → (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝐶)
42, 3syl5eqel 2166 . . 3 ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝑅 × 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)
51, 4sylan2 280 . 2 ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ (𝐴𝑅𝐵𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶)
653impb 1135 1 ((𝐹:(𝑅 × 𝑆)⟶𝐶𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920  wcel 1434  cop 3409   × cxp 4369  wf 4928  cfv 4932  (class class class)co 5543
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fv 4940  df-ov 5546
This theorem is referenced by:  fovrnda  5675  fovrnd  5676
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