Theorem funres11 6593

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 5983	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		cnvss 5840	. 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹)
3		funss 6535	. 2 ⊢ (◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 → (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴)))
4	1, 2, 3	mp2b 10	1 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3902  ^◡ccnv 5642   ↾ cres 5645  Fun wfun 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3909  df-ss 3919  df-br 5098  df-opab 5160  df-rel 5650  df-cnv 5651  df-co 5652  df-res 5655  df-fun 6518

This theorem is referenced by:  f1ssres  6764  resdif  6823  f1ssf1  6834  f1oi  6840  resf1extb  7910  ssdomg  8975  sbthlem8  9060  spthispth  29881