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Theorem ffvelrni 5301
 Description: A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
Hypothesis
Ref Expression
ffvrni.1 𝐹:𝐴𝐵
Assertion
Ref Expression
ffvelrni (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)

Proof of Theorem ffvelrni
StepHypRef Expression
1 ffvrni.1 . 2 𝐹:𝐴𝐵
2 ffvelrn 5300 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)
31, 2mpan 400 1 (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  ⟶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:  cjcl  9448  climmpt  9821  cn1lem  9834  climcn1lem  9839  ialgcvg  9887  algcvgb  9889  ialgcvga  9890  ialgfx  9891
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