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Theorem fss 5054
 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 2952 . . . . 5 (ran 𝐹𝐵 → (𝐵𝐶 → ran 𝐹𝐶))
21com12 27 . . . 4 (𝐵𝐶 → (ran 𝐹𝐵 → ran 𝐹𝐶))
32anim2d 320 . . 3 (𝐵𝐶 → ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
4 df-f 4906 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
5 df-f 4906 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
63, 4, 53imtr4g 194 . 2 (𝐵𝐶 → (𝐹:𝐴𝐵𝐹:𝐴𝐶))
76impcom 116 1 ((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ⊆ wss 2917  ran crn 4346   Fn wfn 4897  ⟶wf 4898 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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931  df-f 4906 This theorem is referenced by:  fssd  5055  fconst6g  5085  f1ss  5097  ffoss  5158  fsn2  5337  ofco  5729  tposf2  5883  issmo2  5904  smoiso  5917  ssdomg  6258  1fv  8996  abscn2  9835  recn2  9837  imcn2  9838  climabs  9840  climre  9842  climim  9843
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