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Theorem ffvelrn 5225
Description: A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
Assertion
Ref Expression
ffvelrn ((𝐹:AB 𝐶 A) → (𝐹𝐶) B)

Proof of Theorem ffvelrn
StepHypRef Expression
1 ffn 4972 . . 3 (𝐹:AB𝐹 Fn A)
2 fnfvelrn 5224 . . 3 ((𝐹 Fn A 𝐶 A) → (𝐹𝐶) ran 𝐹)
31, 2sylan 267 . 2 ((𝐹:AB 𝐶 A) → (𝐹𝐶) ran 𝐹)
4 frn 4978 . . . 4 (𝐹:AB → ran 𝐹B)
54sseld 2921 . . 3 (𝐹:AB → ((𝐹𝐶) ran 𝐹 → (𝐹𝐶) B))
65adantr 261 . 2 ((𝐹:AB 𝐶 A) → ((𝐹𝐶) ran 𝐹 → (𝐹𝐶) B))
73, 6mpd 13 1 ((𝐹:AB 𝐶 A) → (𝐹𝐶) B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1374  ran crn 4273   Fn wfn 4824  wf 4825  cfv 4829
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-f 4833  df-fv 4837
This theorem is referenced by:  ffvelrni  5226  ffvelrnda  5227  dffo3  5239  foco2  5243  ffnfv  5248  ffvresb  5253  fcompt  5258  fsn2  5262  fvconst  5276  fcofo  5349  cocan1  5352  isocnv  5376  isores2  5378  isopolem  5386  isosolem  5388  fovrn  5566  off  5647
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