Theorem foco 5116

Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.)

Assertion

Ref	Expression
foco	|- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( F o. G ) : A -onto-> C )

Proof of Theorem foco

Step	Hyp	Ref	Expression
1		dffo2 5110	. . 3 |- ( F : B -onto-> C <-> ( F : B --> C /\ ran F = C ) )
2		dffo2 5110	. . 3 |- ( G : A -onto-> B <-> ( G : A --> B /\ ran G = B ) )
3		fco 5056	. . . . 5 |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )
4	3	ad2ant2r 478	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ( F o. G ) : A --> C )
5		fdm 5050	. . . . . . . 8 |- ( F : B --> C -> dom F = B )
6		eqtr3 2059	. . . . . . . 8 |- ( ( dom F = B /\ ran G = B ) -> dom F = ran G )
7	5, 6	sylan 267	. . . . . . 7 |- ( ( F : B --> C /\ ran G = B ) -> dom F = ran G )
8		rncoeq 4605	. . . . . . . . 9 |- ( dom F = ran G -> ran ( F o. G ) = ran F )
9	8	eqeq1d 2048	. . . . . . . 8 |- ( dom F = ran G -> ( ran ( F o. G ) = C <-> ran F = C ) )
10	9	biimpar 281	. . . . . . 7 |- ( ( dom F = ran G /\ ran F = C ) -> ran ( F o. G ) = C )
11	7, 10	sylan 267	. . . . . 6 |- ( ( ( F : B --> C /\ ran G = B ) /\ ran F = C ) -> ran ( F o. G ) = C )
12	11	an32s 502	. . . . 5 |- ( ( ( F : B --> C /\ ran F = C ) /\ ran G = B ) -> ran ( F o. G ) = C )
13	12	adantrl 447	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ran ( F o. G ) = C )
14	4, 13	jca 290	. . 3 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
15	1, 2, 14	syl2anb 275	. 2 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
16		dffo2 5110	. 2 |- ( ( F o. G ) : A -onto-> C <-> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
17	15, 16	sylibr 137	1 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( F o. G ) : A -onto-> C )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   dom cdm 4345   ran crn 4346   o. ccom 4349   -->wf 4898   -onto->wfo 4900

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908

This theorem is referenced by:  f1oco  5149