Theorem funres11 6617

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 6002	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		cnvss 5860	. 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹)
3		funss 6559	. 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 3906  ^◡ccnv 5662   ↾ cres 5665  Fun wfun 6534

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3913  df-ss 3923  df-br 5112  df-opab 5176  df-rel 5670  df-cnv 5671  df-co 5672  df-res 5675  df-fun 6542

This theorem is used by:  f1ssres  6787  resdif  6846  f1ssf1  6857  f1oi  6863  resf1extb  7933  ssdomg  8999  sbthlem8  9085  spthispth  30112