Theorem f1ss 5263

Description: A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)

Assertion

Ref	Expression
f1ss	⊢ ((F:A–1-1→B ∧ B ⊆ C) → F:A–1-1→C)

Proof of Theorem f1ss

Step	Hyp	Ref	Expression
1		f1f 5259	. . 3 ⊢ (F:A–1-1→B → F:A–→B)
2		fss 5231	. . 3 ⊢ ((F:A–→B ∧ B ⊆ C) → F:A–→C)
3	1, 2	sylan 457	. 2 ⊢ ((F:A–1-1→B ∧ B ⊆ C) → F:A–→C)
4		df-f1 4793	. . . 4 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ Fun ◡F))
5	4	simprbi 450	. . 3 ⊢ (F:A–1-1→B → Fun ◡F)
6	5	adantr 451	. 2 ⊢ ((F:A–1-1→B ∧ B ⊆ C) → Fun ◡F)
7		df-f1 4793	. 2 ⊢ (F:A–1-1→C ↔ (F:A–→C ∧ Fun ◡F))
8	3, 6, 7	sylanbrc 645	1 ⊢ ((F:A–1-1→B ∧ B ⊆ C) → F:A–1-1→C)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ⊆ wss 3258  ^◡ccnv 4772  Fun wfun 4776  –→wf 4778  –1-1→wf1 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 proof 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-f 4792  df-f1 4793

This theorem is used by:  dflec3  6222