Theorem fnovrn 7542

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 5668	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2		df-ov 7370	. . . 4 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
3		fnfvelrn 7032	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
4	2, 3	eqeltrid 2840	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
5	1, 4	sylan2 594	. 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 2114  ⟨cop 4573   × cxp 5629  ran crn 5632   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506  df-ov 7370

This theorem is referenced by:  unirnioo  13402  ioorebas  13404  yonffthlem  18248  gsumval2a  18653  efginvrel2  19702  efgredleme  19718  efgcpbllemb  19730  mplsubrglem  21982  lecldbas  23184  blelrnps  24381  blelrn  24382  blssioo  24760  tgioo  24761  opnmbllem  25568  mbfdm  25593  mbfima  25597  tpr2rico  34056  dya2icoseg  34421  opnmbllem0  37977  elrnmpoid  45657  smflimlem3  47201