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Theorem fss 5284
Description: Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Assertion
Ref Expression
fss ((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)

Proof of Theorem fss
StepHypRef Expression
1 sstr2 3104 . . . . 5 (ran 𝐹𝐵 → (𝐵𝐶 → ran 𝐹𝐶))
21com12 30 . . . 4 (𝐵𝐶 → (ran 𝐹𝐵 → ran 𝐹𝐶))
32anim2d 335 . . 3 (𝐵𝐶 → ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
4 df-f 5127 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
5 df-f 5127 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
63, 4, 53imtr4g 204 . 2 (𝐵𝐶 → (𝐹:𝐴𝐵𝐹:𝐴𝐶))
76impcom 124 1 ((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wss 3071  ran crn 4540   Fn wfn 5118  wf 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-f 5127
This theorem is referenced by:  fssd  5285  fconst6g  5321  f1ss  5334  ffoss  5399  fsn2  5594  ofco  6000  tposf2  6165  issmo2  6186  smoiso  6199  mapsn  6584  ssdomg  6672  omp1eomlem  6979  1fv  9928  fxnn0nninf  10223  abscn2  11096  recn2  11098  imcn2  11099  climabs  11101  climre  11103  climim  11104  fsumre  11253  fsumim  11254  ismet2  12537  dvfre  12857  dvrecap  12860
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