Theorem fss 5230

Description: Expanding the codomain of a mapping. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 10-May-1998.) (Revised by set.mm contributors, 18-Sep-2011.)

Assertion

Ref	Expression
fss	⊢ ((F:A–→B ∧ B ⊆ C) → F:A–→C)

Proof of Theorem fss

Step	Hyp	Ref	Expression
1		sstr2 3279	. . . . 5 ⊢ (ran F ⊆ B → (B ⊆ C → ran F ⊆ C))
2	1	com12 27	. . . 4 ⊢ (B ⊆ C → (ran F ⊆ B → ran F ⊆ C))
3	2	anim2d 548	. . 3 ⊢ (B ⊆ C → ((F Fn A ∧ ran F ⊆ B) → (F Fn A ∧ ran F ⊆ C)))
4		df-f 4791	. . 3 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
5		df-f 4791	. . 3 ⊢ (F:A–→C ↔ (F Fn A ∧ ran F ⊆ C))
6	3, 4, 5	3imtr4g 261	. 2 ⊢ (B ⊆ C → (F:A–→B → F:A–→C))
7	6	impcom 419	1 ⊢ ((F:A–→B ∧ B ⊆ C) → F:A–→C)

Syntax hints:   → wi 4   ∧ wa 358   ⊆ wss 3257  ran crn 4773   Fn wfn 4776  –→wf 4777

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-f 4791

This theorem is referenced by:  f1ss  5262  ffoss  5314  fsn2  5434  map0  6025  mapsn  6026  mapss  6027