Theorem fnovrn 7543

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 5669	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2		df-ov 7371	. . . 4 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
3		fnfvelrn 7034	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
4	2, 3	eqeltrid 2841	. . 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 4588   × cxp 5630  ran crn 5633   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368

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 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508  df-ov 7371

This theorem is referenced by:  unirnioo  13377  ioorebas  13379  yonffthlem  18217  gsumval2a  18622  efginvrel2  19668  efgredleme  19684  efgcpbllemb  19696  mplsubrglem  21971  lecldbas  23175  blelrnps  24372  blelrn  24373  blssioo  24751  tgioo  24752  opnmbllem  25570  mbfdm  25595  mbfima  25599  tpr2rico  34090  dya2icoseg  34455  opnmbllem0  37907  elrnmpoid  45586  smflimlem3  47131