Theorem funres11 3559

Description: The restriction of a one-to-one function is one-to-one.

Assertion

Ref	Expression
funres11	|- (Fun `'F -> Fun `'(F |` A))

Proof of Theorem funres11

Step	Hyp	Ref	Expression
1		resss 3375	. . 3 |- (F |` A) (_ F
2		cnvss 3286	. . 3 |- ((F |` A) (_ F -> `'(F |` A) (_ `'F)
3	1, 2	ax-mp 7	. 2 |- `'(F |` A) (_ `'F
4		funss 3526	. 2 |- (`'(F |` A) (_ `'F -> (Fun `'F -> Fun `'(F |` A)))
5	3, 4	ax-mp 7	1 |- (Fun `'F -> Fun `'(F |` A))

Syntax hints:   -> wi 3   (_ wss 2043  `'ccnv 3164   |` cres 3167  Fun wfun 3171

This theorem is referenced by:  f1ores 3694  f1oi 3708  ssdomg 4395  sbthlem8 4440

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-rel 3180  df-cnv 3181  df-co 3182  df-res 3185  df-fun 3187

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