Theorem foco 5816

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 5810	. . 3 |- ( F : B -onto-> C <-> ( F : B --> C /\ ran F = C ) )
2		dffo2 5810	. . 3 |- ( G : A -onto-> B <-> ( G : A --> B /\ ran G = B ) )
3		fco 5751	. . . . 5 |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )
4	3	ad2ant2r 761	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ( F o. G ) : A --> C )
5		fdm 5745	. . . . . . . 8 |- ( F : B --> C -> dom F = B )
6		eqtr3 2492	. . . . . . . 8 |- ( ( dom F = B /\ ran G = B ) -> dom F = ran G )
7	5, 6	sylan 479	. . . . . . 7 |- ( ( F : B --> C /\ ran G = B ) -> dom F = ran G )
8		rncoeq 5104	. . . . . . . . 9 |- ( dom F = ran G -> ran ( F o. G ) = ran F )
9	8	eqeq1d 2473	. . . . . . . 8 |- ( dom F = ran G -> ( ran ( F o. G ) = C <-> ran F = C ) )
10	9	biimpar 493	. . . . . . 7 |- ( ( dom F = ran G /\ ran F = C ) -> ran ( F o. G ) = C )
11	7, 10	sylan 479	. . . . . 6 |- ( ( ( F : B --> C /\ ran G = B ) /\ ran F = C ) -> ran ( F o. G ) = C )
12	11	an32s 821	. . . . 5 |- ( ( ( F : B --> C /\ ran F = C ) /\ ran G = B ) -> ran ( F o. G ) = C )
13	12	adantrl 730	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ran ( F o. G ) = C )
14	4, 13	jca 541	. . 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 487	. 2 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
16		dffo2 5810	. 2 |- ( ( F o. G ) : A -onto-> C <-> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
17	15, 16	sylibr 217	1 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( F o. G ) : A -onto-> C )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   dom cdm 4839   ran crn 4840   o. ccom 4843   -->wf 5585   -onto->wfo 5587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-fun 5591  df-fn 5592  df-f 5593  df-fo 5595

This theorem is referenced by:  f1oco  5850  wdomtr  8108  fin1a2lem7  8854  cofull  15917  uniiccdif  22614