Theorem foco 5509

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 5502	. . 3 |- ( F : B -onto-> C <-> ( F : B --> C /\ ran F = C ) )
2		dffo2 5502	. . 3 |- ( G : A -onto-> B <-> ( G : A --> B /\ ran G = B ) )
3		fco 5441	. . . . 5 |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )
4	3	ad2ant2r 509	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ( F o. G ) : A --> C )
5		fdm 5431	. . . . . . . 8 |- ( F : B --> C -> dom F = B )
6		eqtr3 2225	. . . . . . . 8 |- ( ( dom F = B /\ ran G = B ) -> dom F = ran G )
7	5, 6	sylan 283	. . . . . . 7 |- ( ( F : B --> C /\ ran G = B ) -> dom F = ran G )
8		rncoeq 4952	. . . . . . . . 9 |- ( dom F = ran G -> ran ( F o. G ) = ran F )
9	8	eqeq1d 2214	. . . . . . . 8 |- ( dom F = ran G -> ( ran ( F o. G ) = C <-> ran F = C ) )
10	9	biimpar 297	. . . . . . 7 |- ( ( dom F = ran G /\ ran F = C ) -> ran ( F o. G ) = C )
11	7, 10	sylan 283	. . . . . 6 |- ( ( ( F : B --> C /\ ran G = B ) /\ ran F = C ) -> ran ( F o. G ) = C )
12	11	an32s 568	. . . . 5 |- ( ( ( F : B --> C /\ ran F = C ) /\ ran G = B ) -> ran ( F o. G ) = C )
13	12	adantrl 478	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ran ( F o. G ) = C )
14	4, 13	jca 306	. . 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 291	. 2 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
16		dffo2 5502	. 2 |- ( ( F o. G ) : A -onto-> C <-> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
17	15, 16	sylibr 134	1 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( F o. G ) : A -onto-> C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   dom cdm 4675   ran crn 4676   o. ccom 4679   -->wf 5267   -onto->wfo 5269

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277

This theorem is referenced by:  f1oco  5545  nninfct  12362  ennnfonelemnn0  12793  ctinfomlemom  12798  qnnen  12802  enctlem  12803