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Theorem fnovrn 5571
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
StepHypRef Expression
1 opelxpi 4303 . . 3 ((𝐶 A 𝐷 B) → ⟨𝐶, 𝐷 (A × B))
2 df-ov 5439 . . . 4 (𝐶𝐹𝐷) = (𝐹‘⟨𝐶, 𝐷⟩)
3 fnfvelrn 5224 . . . 4 ((𝐹 Fn (A × B) 𝐶, 𝐷 (A × B)) → (𝐹‘⟨𝐶, 𝐷⟩) ran 𝐹)
42, 3syl5eqel 2106 . . 3 ((𝐹 Fn (A × B) 𝐶, 𝐷 (A × B)) → (𝐶𝐹𝐷) ran 𝐹)
51, 4sylan2 270 . 2 ((𝐹 Fn (A × B) (𝐶 A 𝐷 B)) → (𝐶𝐹𝐷) ran 𝐹)
653impb 1086 1 ((𝐹 Fn (A × B) 𝐶 A 𝐷 B) → (𝐶𝐹𝐷) ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 873   wcel 1374  cop 3353   × cxp 4270  ran crn 4273   Fn wfn 4824  cfv 4829  (class class class)co 5436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837  df-ov 5439
This theorem is referenced by: (None)
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