Theorem f1ssf1 5666

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 5415	. . . . 5 |- ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
2		funres11 5448	. . . . . . 7 |- ( Fun `' F -> Fun `' ( F |` dom G ) )
3		cnveq 4949	. . . . . . . 8 |- ( G = ( F |` dom G ) -> `' G = `' ( F |` dom G ) )
4	3	funeqd 5394	. . . . . . 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 2241	. . . . 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 1224	1 |- ( ( Fun F /\ Fun `' F /\ G C_ F ) -> Fun `' G )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   C_ wss 3220   `'ccnv 4768   dom cdm 4769   |` cres 4771   Fun wfun 5366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-fun 5374

This theorem is referenced by:  subusgr  16430