Theorem foco 3696

Description: Composition of onto functions.

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		fco 3650	. . . 4 |- ((F:B-->C /\ G:A-->B) -> (F o. G):A-->C)
2	1	ad2ant2r 411	. . 3 |- (((F:B-->C /\ ran F = C) /\ (G:A-->B /\ ran G = B)) -> (F o. G):A-->C)
3		rncoeq 3381	. . . . . . . 8 |- (dom F = ran G -> ran ( F o. G) = ran F)
4	3	eqeq1d 1490	. . . . . . 7 |- (dom F = ran G -> (ran ( F o. G) = C <-> ran F = C))
5	4	biimpar 419	. . . . . 6 |- ((dom F = ran G /\ ran F = C) -> ran ( F o. G) = C)
6		eqtr3t 1501	. . . . . . 7 |- ((dom F = B /\ ran G = B) -> dom F = ran G)
7		fdm 3645	. . . . . . 7 |- (F:B-->C -> dom F = B)
8	6, 7	sylan 451	. . . . . 6 |- ((F:B-->C /\ ran G = B) -> dom F = ran G)
9	5, 8	sylan 451	. . . . 5 |- (((F:B-->C /\ ran G = B) /\ ran F = C) -> ran ( F o. G) = C)
10	9	an1rs 492	. . . 4 |- (((F:B-->C /\ ran F = C) /\ ran G = B) -> ran ( F o. G) = C)
11	10	adantrl 396	. . 3 |- (((F:B-->C /\ ran F = C) /\ (G:A-->B /\ ran G = B)) -> ran ( F o. G) = C)
12	2, 11	jca 288	. 2 |- (((F:B-->C /\ ran F = C) /\ (G:A-->B /\ ran G = B)) -> ((F o. G):A-->C /\ ran ( F o. G) = C))
13		dffo2 3689	. . 3 |- (F:B-onto->C <-> (F:B-->C /\ ran F = C))
14		dffo2 3689	. . 3 |- (G:A-onto->B <-> (G:A-->B /\ ran G = B))
15	13, 14	anbi12i 485	. 2 |- ((F:B-onto->C /\ G:A-onto->B) <-> ((F:B-->C /\ ran F = C) /\ (G:A-->B /\ ran G = B)))
16		dffo2 3689	. 2 |- ((F o. G):A-onto->C <-> ((F o. G):A-->C /\ ran ( F o. G) = C))
17	12, 15, 16	3imtr4 219	1 |- ((F:B-onto->C /\ G:A-onto->B) -> (F o. G):A-onto->C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 960  dom cdm 3184  ran crn 3185   o. ccom 3188  -->wf 3192  -onto->wfo 3194

This theorem is referenced by:  f1oco 3721  fodomfi 4576

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-br 2633  df-opab 2680  df-id 2849  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-fun 3206  df-fn 3207  df-f 3208  df-fo 3210

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