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Theorem fss 5378
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 3163 . . . . 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 5221 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
5 df-f 5221 . . 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 3130  ran crn 4628   Fn wfn 5212  wf 5213
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 3136  df-ss 3143  df-f 5221
This theorem is referenced by:  fssd  5379  fconst6g  5415  f1ss  5428  ffoss  5494  fsn2  5691  ofco  6101  tposf2  6269  issmo2  6290  smoiso  6303  mapsn  6690  ssdomg  6778  omp1eomlem  7093  1fv  10139  fxnn0nninf  10438  abscn2  11323  recn2  11325  imcn2  11326  climabs  11328  climre  11330  climim  11331  fsumre  11480  fsumim  11481  ismet2  13857  dvfre  14177  dvrecap  14180  lgsfcl  14412
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