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Theorem ffvelrn 5243
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 4989 . . 3 (𝐹:AB𝐹 Fn A)
2 fnfvelrn 5242 . . 3 ((𝐹 Fn A 𝐶 A) → (𝐹𝐶) ran 𝐹)
31, 2sylan 267 . 2 ((𝐹:AB 𝐶 A) → (𝐹𝐶) ran 𝐹)
4 frn 4995 . . . 4 (𝐹:AB → ran 𝐹B)
54sseld 2938 . . 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 1390  ran crn 4289   Fn wfn 4840  wf 4841  cfv 4845
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853
This theorem is referenced by:  ffvelrni  5244  ffvelrnda  5245  dffo3  5257  foco2  5261  ffnfv  5266  ffvresb  5271  fcompt  5276  fsn2  5280  fvconst  5294  fcofo  5367  cocan1  5370  isocnv  5394  isores2  5396  isopolem  5404  isosolem  5406  fovrn  5585  off  5666  2dom  6221  enm  6230  xpdom2  6241
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