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Theorem fss 5377
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 3162 . . . . 5 (ran 𝐹𝐵 → (𝐵𝐶 → ran 𝐹𝐶))
21com12 30 . . . 4 (𝐵𝐶 → (ran 𝐹𝐵 → ran 𝐹𝐶))
32anim2d 337 . . 3 (𝐵𝐶 → ((𝐹 Fn 𝐴 ∧ ran 𝐹𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶)))
4 df-f 5220 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
5 df-f 5220 . . 3 (𝐹:𝐴𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐶))
63, 4, 53imtr4g 205 . 2 (𝐵𝐶 → (𝐹:𝐴𝐵𝐹:𝐴𝐶))
76impcom 125 1 ((𝐹:𝐴𝐵𝐵𝐶) → 𝐹:𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wss 3129  ran crn 4627   Fn wfn 5211  wf 5212
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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142  df-f 5220
This theorem is referenced by:  fssd  5378  fconst6g  5414  f1ss  5427  ffoss  5493  fsn2  5690  ofco  6100  tposf2  6268  issmo2  6289  smoiso  6302  mapsn  6689  ssdomg  6777  omp1eomlem  7092  1fv  10138  fxnn0nninf  10437  abscn2  11322  recn2  11324  imcn2  11325  climabs  11327  climre  11329  climim  11330  fsumre  11479  fsumim  11480  ismet2  13824  dvfre  14144  dvrecap  14147  lgsfcl  14379
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