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Theorem fnovrn 6956
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 5288 . . 3 ((𝐶𝐴𝐷𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2 df-ov 6796 . . . 4 (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
3 fnfvelrn 6499 . . . 4 ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
42, 3syl5eqel 2854 . . 3 ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
51, 4sylan2 580 . 2 ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶𝐴𝐷𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
653impb 1107 1 ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐴𝐷𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071  wcel 2145  cop 4322   × cxp 5247  ran crn 5250   Fn wfn 6026  cfv 6031  (class class class)co 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ov 6796
This theorem is referenced by:  unirnioo  12479  ioorebas  12481  yonffthlem  17130  gsumval2a  17487  efginvrel2  18347  efgredleme  18363  efgcpbllemb  18375  mplsubrglem  19654  lecldbas  21244  blelrnps  22441  blelrn  22442  blssioo  22818  tgioo  22819  opnmbllem  23589  mbfdm  23614  mbfima  23618  tpr2rico  30298  dya2icoseg  30679  opnmbllem0  33778  elrnmpt2id  39945  smflimlem3  41501
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