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Theorem fss 5105
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 3015 . . . . 5 (ran 𝐹𝐵 → (𝐵𝐶 → ran 𝐹𝐶))
21com12 30 . . . 4 (𝐵𝐶 → (ran 𝐹𝐵 → ran 𝐹𝐶))
32anim2d 330 . . 3 (𝐵𝐶 → ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
4 df-f 4956 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
5 df-f 4956 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
63, 4, 53imtr4g 203 . 2 (𝐵𝐶 → (𝐹:𝐴𝐵𝐹:𝐴𝐶))
76impcom 123 1 ((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wss 2982  ran crn 4392   Fn wfn 4947  wf 4948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2988  df-ss 2995  df-f 4956
This theorem is referenced by:  fssd  5106  fconst6g  5136  f1ss  5148  ffoss  5209  fsn2  5389  ofco  5780  tposf2  5937  issmo2  5958  smoiso  5971  ssdomg  6346  1fv  9278  abscn2  10354  recn2  10356  imcn2  10357  climabs  10359  climre  10361  climim  10362
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