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Theorem fnovrn 6763
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 5113 . . 3 ((𝐶𝐴𝐷𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2 df-ov 6608 . . . 4 (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
3 fnfvelrn 6313 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
42, 3syl5eqel 2708 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
51, 4sylan2 491 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
653impb 1257 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wcel 1992  cop 4159   × cxp 5077  ran crn 5080   Fn wfn 5845  cfv 5850  (class class class)co 6605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-fv 5858  df-ov 6608
This theorem is referenced by:  unirnioo  12212  ioorebas  12214  yonffthlem  16838  gsumval2a  17195  efginvrel2  18056  efgredleme  18072  efgcpbllemb  18084  mplsubrglem  19353  lecldbas  20928  blelrnps  22126  blelrn  22127  blssioo  22501  tgioo  22502  opnmbllem  23270  mbfdm  23296  mbfima  23300  tpr2rico  29732  dya2icoseg  30112  opnmbllem0  33063  elrnmpt2id  38887  smflimlem3  40275
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