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Theorem fss 5157
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 3030 . . . . 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 5006 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
5 df-f 5006 . . 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 2997  ran crn 4429   Fn wfn 4997  wf 4998
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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-f 5006
This theorem is referenced by:  fssd  5158  fconst6g  5193  f1ss  5206  ffoss  5269  fsn2  5455  ofco  5855  tposf2  6015  issmo2  6036  smoiso  6049  mapsn  6427  ssdomg  6475  1fv  9515  fxnn0nninf  9809  abscn2  10667  recn2  10669  imcn2  10670  climabs  10672  climre  10674  climim  10675  fsumre  10829  fsumim  10830
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