Theorem funres11 5165

Description: The restriction of a one-to-one function is one-to-one. (Contributed by set.mm contributors, 25-Mar-1998.)

Assertion

Ref	Expression
funres11	⊢ (Fun ◡F → Fun ◡(F ↾ A))

Proof of Theorem funres11

Step	Hyp	Ref	Expression
1		resss 4989	. 2 ⊢ (F ↾ A) ⊆ F
2		cnvss 4886	. 2 ⊢ ((F ↾ A) ⊆ F → ◡(F ↾ A) ⊆ ◡F)
3		funss 5127	. 2 ⊢ (◡(F ↾ A) ⊆ ◡F → (Fun ◡F → Fun ◡(F ↾ A)))
4	1, 2, 3	mp2b 9	1 ⊢ (Fun ◡F → Fun ◡(F ↾ A))

Syntax hints:   → wi 4   ⊆ wss 3258  ^◡ccnv 4772   ↾ cres 4775  Fun wfun 4776

This theorem was proved from 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 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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727  df-cnv 4786  df-res 4789  df-fun 4790

This theorem is referenced by:  f1ores  5301  resdif  5307