ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ffvelcdmi GIF version

Theorem ffvelcdmi 5645
Description: A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
Hypothesis
Ref Expression
ffvelcdmi.1 𝐹:𝐴𝐵
Assertion
Ref Expression
ffvelcdmi (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)

Proof of Theorem ffvelcdmi
StepHypRef Expression
1 ffvelcdmi.1 . 2 𝐹:𝐴𝐵
2 ffvelcdm 5644 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)
31, 2mpan 424 1 (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  wf 5207  cfv 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219
This theorem is referenced by:  omgadd  10753  cjcl  10828  climmpt  11279  cn1lem  11293  climcn1lem  11298  fsumrelem  11450  efcl  11643  sincl  11685  coscl  11686  algcvg  12018  algcvgb  12020  algcvga  12021  algfx  12022  eucalgcvga  12028  eucalg  12029  sqpweven  12145  2sqpwodd  12146  ennnfonelemnn0  12393  relogcl  13916  nninfomnilem  14390
  Copyright terms: Public domain W3C validator