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Theorem ffvelrn 5300
Description: A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
Assertion
Ref Expression
ffvelrn ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)

Proof of Theorem ffvelrn
StepHypRef Expression
1 ffn 5046 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fnfvelrn 5299 . . 3 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) ∈ ran 𝐹)
31, 2sylan 267 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ ran 𝐹)
4 frn 5052 . . . 4 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
54sseld 2944 . . 3 (𝐹:𝐴𝐵 → ((𝐹𝐶) ∈ ran 𝐹 → (𝐹𝐶) ∈ 𝐵))
65adantr 261 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ((𝐹𝐶) ∈ ran 𝐹 → (𝐹𝐶) ∈ 𝐵))
73, 6mpd 13 1 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wcel 1393  ran crn 4346   Fn wfn 4897  wf 4898  cfv 4902
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910
This theorem is referenced by:  ffvelrni  5301  ffvelrnda  5302  dffo3  5314  foco2  5318  ffnfv  5323  ffvresb  5328  fcompt  5333  fsn2  5337  fvconst  5351  fcofo  5424  cocan1  5427  isocnv  5451  isores2  5453  isopolem  5461  isosolem  5463  fovrn  5643  off  5724  2dom  6285  enm  6294  xpdom2  6305  shftf  9431  nn0seqcvgd  9880
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