Theorem funres11 5260

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 4908	. 2 ⊢ (𝐹 ↾ 𝐴) ⊆ 𝐹
2		cnvss 4777	. 2 ⊢ ((𝐹 ↾ 𝐴) ⊆ 𝐹 → ◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹)
3		funss 5207	. 2 ⊢ (◡(𝐹 ↾ 𝐴) ⊆ ◡𝐹 → (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴)))
4	1, 2, 3	mp2b 8	1 ⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴))

Syntax hints:   → wi 4   ⊆ wss 3116  ^◡ccnv 4603   ↾ cres 4606  Fun wfun 5182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-br 3983  df-opab 4044  df-rel 4611  df-cnv 4612  df-co 4613  df-res 4616  df-fun 5190

This theorem is referenced by:  f1ssres  5402  resdif  5454  ssdomg  6744  sbthlemi8  6929