Theorem fnovrn 7608

Description: An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.)

Assertion

Ref	Expression
fnovrn	⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)

Proof of Theorem fnovrn

Step	Hyp	Ref	Expression
1		opelxpi 5722	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2		df-ov 7434	. . . 4 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
3		fnfvelrn 7100	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
4	2, 3	eqeltrid 2845	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
5	1, 4	sylan2 593	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
6	5	3impb 1115	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ⟨cop 4632   × cxp 5683  ran crn 5686   Fn wfn 6556  ‘cfv 6561  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434

This theorem is referenced by:  unirnioo  13489  ioorebas  13491  yonffthlem  18327  gsumval2a  18698  efginvrel2  19745  efgredleme  19761  efgcpbllemb  19773  mplsubrglem  22024  lecldbas  23227  blelrnps  24426  blelrn  24427  blssioo  24816  tgioo  24817  opnmbllem  25636  mbfdm  25661  mbfima  25665  tpr2rico  33911  dya2icoseg  34279  opnmbllem0  37663  elrnmpoid  45233  smflimlem3  46788