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Theorem fnovrn 7602
Description: An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)
Assertion
Ref Expression
fnovrn ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)

Proof of Theorem fnovrn
StepHypRef Expression
1 opelxpi 5719 . . 3 ((𝐶𝐴𝐷𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2 df-ov 7429 . . . 4 (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
3 fnfvelrn 7095 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
42, 3eqeltrid 2833 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
51, 4sylan2 591 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
653impb 1112 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084  wcel 2098  cop 4638   × cxp 5680  ran crn 5683   Fn wfn 6548  cfv 6553  (class class class)co 7426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-fv 6561  df-ov 7429
This theorem is referenced by:  unirnioo  13466  ioorebas  13468  yonffthlem  18281  gsumval2a  18652  efginvrel2  19689  efgredleme  19705  efgcpbllemb  19717  mplsubrglem  21953  lecldbas  23143  blelrnps  24342  blelrn  24343  blssioo  24731  tgioo  24732  opnmbllem  25550  mbfdm  25575  mbfima  25579  tpr2rico  33546  dya2icoseg  33930  opnmbllem0  37162  elrnmpoid  44631  smflimlem3  46190
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