Theorem funres11 6620

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

Assertion

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

Proof of Theorem funres11

Step	Hyp	Ref	Expression
1		resss 5992	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		cnvss 5850	. 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹)
3		funss 6561	. 2 ⊢ (◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 → (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴)))
4	1, 2, 3	mp2b 10	1 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3896  ^◡ccnv 5649   ↾ cres 5652  Fun wfun 6535

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3897  df-in 4090  df-br 5103  df-opab 5167  df-rel 5657  df-cnv 5658  df-co 5659  df-res 5662  df-fun 6543

This theorem is used by:  f1ssres  6790  resdif  6849  f1ssf1  6860  f1oi  6866  resf1extb  7948  ssdomg  9032  sbthlem8  9118  spthispth  30316