Theorem funres11 6645

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 6022	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		cnvss 5886	. 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹)
3		funss 6587	. 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 3963  ^◡ccnv 5688   ↾ cres 5691  Fun wfun 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-ss 3980  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-res 5701  df-fun 6565

This theorem is referenced by:  f1ssres  6812  resdif  6870  f1ssf1  6881  ssdomg  9039  sbthlem8  9129  spthispth  29759