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Theorem f1ssf1 39062
Description: A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
Assertion
Ref Expression
f1ssf1  |-  ( ( Fun  F  /\  Fun  `' F  /\  G  C_  F )  ->  Fun  `' G )

Proof of Theorem f1ssf1
StepHypRef Expression
1 funssres 5640 . . . . 5  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( F  |`  dom  G )  =  G )
2 funres11 5672 . . . . . . 7  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  dom  G
) )
3 cnveq 5026 . . . . . . . 8  |-  ( G  =  ( F  |`  dom  G )  ->  `' G  =  `' ( F  |`  dom  G ) )
43funeqd 5621 . . . . . . 7  |-  ( G  =  ( F  |`  dom  G )  ->  ( Fun  `' G  <->  Fun  `' ( F  |`  dom  G ) ) )
52, 4syl5ibr 229 . . . . . 6  |-  ( G  =  ( F  |`  dom  G )  ->  ( Fun  `' F  ->  Fun  `' G ) )
65eqcoms 2469 . . . . 5  |-  ( ( F  |`  dom  G )  =  G  ->  ( Fun  `' F  ->  Fun  `' G ) )
71, 6syl 17 . . . 4  |-  ( ( Fun  F  /\  G  C_  F )  ->  ( Fun  `' F  ->  Fun  `' G ) )
87ex 440 . . 3  |-  ( Fun 
F  ->  ( G  C_  F  ->  ( Fun  `' F  ->  Fun  `' G
) ) )
98com23 81 . 2  |-  ( Fun 
F  ->  ( Fun  `' F  ->  ( G  C_  F  ->  Fun  `' G
) ) )
1093imp 1208 1  |-  ( ( Fun  F  /\  Fun  `' F  /\  G  C_  F )  ->  Fun  `' G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    /\ w3a 991    = wceq 1454    C_ wss 3415   `'ccnv 4851   dom cdm 4852    |` cres 4854   Fun wfun 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-res 4864  df-fun 5602
This theorem is referenced by:  subusgr  39410
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