Theorem foco 5817

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 5811	. . 3 |- ( F : B -onto-> C <-> ( F : B --> C /\ ran F = C ) )
2		dffo2 5811	. . 3 |- ( G : A -onto-> B <-> ( G : A --> B /\ ran G = B ) )
3		fco 5753	. . . . 5 |- ( ( F : B --> C /\ G : A --> B ) -> ( F o. G ) : A --> C )
4	3	ad2ant2r 751	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ( F o. G ) : A --> C )
5		fdm 5747	. . . . . . . 8 |- ( F : B --> C -> dom F = B )
6		eqtr3 2450	. . . . . . . 8 |- ( ( dom F = B /\ ran G = B ) -> dom F = ran G )
7	5, 6	sylan 473	. . . . . . 7 |- ( ( F : B --> C /\ ran G = B ) -> dom F = ran G )
8		rncoeq 5114	. . . . . . . . 9 |- ( dom F = ran G -> ran ( F o. G ) = ran F )
9	8	eqeq1d 2424	. . . . . . . 8 |- ( dom F = ran G -> ( ran ( F o. G ) = C <-> ran F = C ) )
10	9	biimpar 487	. . . . . . 7 |- ( ( dom F = ran G /\ ran F = C ) -> ran ( F o. G ) = C )
11	7, 10	sylan 473	. . . . . 6 |- ( ( ( F : B --> C /\ ran G = B ) /\ ran F = C ) -> ran ( F o. G ) = C )
12	11	an32s 811	. . . . 5 |- ( ( ( F : B --> C /\ ran F = C ) /\ ran G = B ) -> ran ( F o. G ) = C )
13	12	adantrl 720	. . . 4 |- ( ( ( F : B --> C /\ ran F = C ) /\ ( G : A --> B /\ ran G = B ) ) -> ran ( F o. G ) = C )
14	4, 13	jca 534	. . 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 481	. 2 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
16		dffo2 5811	. 2 |- ( ( F o. G ) : A -onto-> C <-> ( ( F o. G ) : A --> C /\ ran ( F o. G ) = C ) )
17	15, 16	sylibr 215	1 |- ( ( F : B -onto-> C /\ G : A -onto-> B ) -> ( F o. G ) : A -onto-> C )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   dom cdm 4850   ran crn 4851   o. ccom 4854   -->wf 5594   -onto->wfo 5596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-opab 4480  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-fun 5600  df-fn 5601  df-f 5602  df-fo 5604

This theorem is referenced by:  f1oco  5850  wdomtr  8093  fin1a2lem7  8837  cofull  15827  uniiccdif  22522