Theorem f1ssf1 5651

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

Step	Hyp	Ref	Expression
1		funssres 5400	. . . . 5 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
2		funres11 5433	. . . . . . 7 |- ( Fun `' F -> Fun `' ( F |` dom G ) )
3		cnveq 4934	. . . . . . . 8 |- ( G = ( F |` dom G ) -> `' G = `' ( F |` dom G ) )
4	3	funeqd 5379	. . . . . . 7 |- ( G = ( F |` dom G ) -> ( Fun `' G <-> Fun `' ( F |` dom G ) ) )
5	2, 4	imbitrrid 156	. . . . . 6 |- ( G = ( F |` dom G ) -> ( Fun `' F -> Fun `' G ) )
6	5	eqcoms 2237	. . . . 5 |- ( ( F |` dom G ) = G -> ( Fun `' F -> Fun `' G ) )
7	1, 6	syl 14	. . . 4 |- ( ( Fun F /\ G C_ F ) -> ( Fun `' F -> Fun `' G ) )
8	7	ex 115	. . 3 |- ( Fun F -> ( G C_ F -> ( Fun `' F -> Fun `' G ) ) )
9	8	com23 78	. 2 |- ( Fun F -> ( Fun `' F -> ( G C_ F -> Fun `' G ) ) )
10	9	3imp 1220	1 |- ( ( Fun F /\ Fun `' F /\ G C_ F ) -> Fun `' G )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   C_ wss 3214   `'ccnv 4753   dom cdm 4754   |` cres 4756   Fun wfun 5351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-res 4766  df-fun 5359

This theorem is referenced by:  subusgr  16396