Theorem fss 5544

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:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶)

Proof of Theorem fss

Step	Hyp	Ref	Expression
1		sstr2 3255	. . . . 5 ⊢ (ran 𝐹 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → ran 𝐹 ⊆ 𝐶))
2	1	com12 30	. . . 4 ⊢ (𝐵 ⊆ 𝐶 → (ran 𝐹 ⊆ 𝐵 → ran 𝐹 ⊆ 𝐶))
3	2	anim2d 337	. . 3 ⊢ (𝐵 ⊆ 𝐶 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶)))
4		df-f 5379	. . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵))
5		df-f 5379	. . 3 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶))
6	3, 4, 5	3imtr4g 205	. 2 ⊢ (𝐵 ⊆ 𝐶 → (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶𝐶))
7	6	impcom 125	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ⊆ wss 3220  ran crn 4773   Fn wfn 5370  ⟶wf 5371

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 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-in 3226  df-ss 3233  df-f 5379

This theorem is referenced by:  fssd  5545  fconst6g  5589  f1ss  5602  ffoss  5670  fsn2  5876  ofco  6315  tposf2  6533  issmo2  6554  smoiso  6567  mapsn  6966  ssdomg  7059  omp1eomlem  7428  1fv  10529  fxnn0nninf  10859  hashf1lem1  11268  abscn2  12064  recn2  12066  imcn2  12067  climabs  12069  climre  12071  climim  12072  fsumre  12222  fsumim  12223  resmhm2  13778  prdsgrpd  14180  prdsinvgd  14181  ismet2  15438  dvfre  15794  dvrecap  15797  elplyr  15824  lgsfcl  16110  konigsbergssiedgwen  16710