Theorem fnovrn 7571

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 5684	. . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵))
2		df-ov 7399	. . . 4 ⊢ (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
3		fnfvelrn 7061	. . . 4 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐹‘⟨𝐶, 𝐷⟩) ∈ ran 𝐹)
4	2, 3	eqeltrid 2866	. . 3 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ ⟨𝐶, 𝐷⟩ ∈ (𝐴 × 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
5	1, 4	sylan2 602	. 2 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) → (𝐶𝐹𝐷) ∈ ran 𝐹)
6	5	3impb 1127	1 ⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   ∈ wcel 2142  ⟨cop 4588   × cxp 5645  ran crn 5648   Fn wfn 6516  ‘cfv 6521  (class class class)co 7396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529  df-ov 7399

This theorem is referenced by:  unirnioo  13453  ioorebas  13455  yonffthlem  18314  gsumval2a  18719  efginvrel2  19767  efgredleme  19783  efgcpbllemb  19795  mplsubrglem  22052  lecldbas  23276  blelrnps  24473  blelrn  24474  blssioo  24852  tgioo  24853  opnmbllem  25660  mbfdm  25685  mbfima  25689  tpr2rico  34206  dya2icoseg  34571  opnmbllem0  38152  elrnmpoid  45800  smflimlem3  47344