Theorem f1ssf1 5615

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 5369	. . . . 5 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
2		funres11 5402	. . . . . . 7 |- ( Fun `' F -> Fun `' ( F |` dom G ) )
3		cnveq 4904	. . . . . . . 8 |- ( G = ( F |` dom G ) -> `' G = `' ( F |` dom G ) )
4	3	funeqd 5348	. . . . . . 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 2234	. . . . 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 1219	1 |- ( ( Fun F /\ Fun `' F /\ G C_ F ) -> Fun `' G )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   C_ wss 3200   `'ccnv 4724   dom cdm 4725   |` cres 4727   Fun wfun 5320

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-fun 5328

This theorem is referenced by:  subusgr  16125