Theorem foco 5420

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 5414	. . 3 |- ( F : B -onto-> C <-> ( F : B --> C /\ ran F = C ) )
2		dffo2 5414	. . 3 |- ( G : A -onto-> B <-> ( G : A --> B /\ ran G = B ) )
3		fco 5353	. . . . 5 |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )
4	3	ad2ant2r 501	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ( F o. G ) : A --> C )
5		fdm 5343	. . . . . . . 8 |- ( F : B --> C -> dom F = B )
6		eqtr3 2185	. . . . . . . 8 |- ( ( dom F = B /\ ran G = B ) -> dom F = ran G )
7	5, 6	sylan 281	. . . . . . 7 |- ( ( F : B --> C /\ ran G = B ) -> dom F = ran G )
8		rncoeq 4877	. . . . . . . . 9 |- ( dom F = ran G -> ran ( F o. G ) = ran F )
9	8	eqeq1d 2174	. . . . . . . 8 |- ( dom F = ran G -> ( ran ( F o. G ) = C <-> ran F = C ) )
10	9	biimpar 295	. . . . . . 7 |- ( ( dom F = ran G /\ ran F = C ) -> ran ( F o. G ) = C )
11	7, 10	sylan 281	. . . . . 6 |- ( ( ( F : B --> C /\ ran G = B ) /\ ran F = C ) -> ran ( F o. G ) = C )
12	11	an32s 558	. . . . 5 |- ( ( ( F : B --> C /\ ran F = C ) /\ ran G = B ) -> ran ( F o. G ) = C )
13	12	adantrl 470	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ran ( F o. G ) = C )
14	4, 13	jca 304	. . 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 289	. 2 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
16		dffo2 5414	. 2 |- ( ( F o. G ) : A -onto-> C <-> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
17	15, 16	sylibr 133	1 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( F o. G ) : A -onto-> C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   dom cdm 4604   ran crn 4605   o. ccom 4608   -->wf 5184   -onto->wfo 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194

This theorem is referenced by:  f1oco  5455  ennnfonelemnn0  12355  ctinfomlemom  12360  qnnen  12364  enctlem  12365