Theorem funres11 6610

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 5994	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		cnvss 5852	. 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹)
3		funss 6552	. 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 3899  ^◡ccnv 5654   ↾ cres 5657  Fun wfun 6527

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-in 3906  df-ss 3916  df-br 5104  df-opab 5168  df-rel 5662  df-cnv 5663  df-co 5664  df-res 5667  df-fun 6535

This theorem is used by:  f1ssres  6780  resdif  6839  f1ssf1  6850  f1oi  6856  resf1extb  7931  ssdomg  9006  sbthlem8  9092  spthispth  30188