Theorem foco 3673

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 3627	. . . 4 |- ((F:B-->C /\ G:A-->B) -> (F o. G):A-->C)
2	1	ad2ant2r 409	. . 3 |- (((F:B-->C /\ ran F = C) /\ (G:A-->B /\ ran G = B)) -> (F o. G):A-->C)
3		rncoeq 3359	. . . . . . . 8 |- (dom F = ran G -> ran ( F o. G) = ran F)
4	3	eqeq1d 1480	. . . . . . 7 |- (dom F = ran G -> (ran ( F o. G) = C <-> ran F = C))
5	4	biimpar 417	. . . . . 6 |- ((dom F = ran G /\ ran F = C) -> ran ( F o. G) = C)
6		eqtr3t 1491	. . . . . . 7 |- ((dom F = B /\ ran G = B) -> dom F = ran G)
7		fdm 3623	. . . . . . 7 |- (F:B-->C -> dom F = B)
8	6, 7	sylan 448	. . . . . 6 |- ((F:B-->C /\ ran G = B) -> dom F = ran G)
9	5, 8	sylan 448	. . . . 5 |- (((F:B-->C /\ ran G = B) /\ ran F = C) -> ran ( F o. G) = C)
10	9	an1rs 489	. . . 4 |- (((F:B-->C /\ ran F = C) /\ ran G = B) -> ran ( F o. G) = C)
11	10	adantrl 394	. . 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 3666	. . 3 |- (F:B-onto->C <-> (F:B-->C /\ ran F = C))
14		dffo2 3666	. . 3 |- (G:A-onto->B <-> (G:A-->B /\ ran G = B))
15	13, 14	anbi12i 482	. 2 |- ((F:B-onto->C /\ G:A-onto->B) <-> ((F:B-->C /\ ran F = C) /\ (G:A-->B /\ ran G = B)))
16		dffo2 3666	. 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 954  dom cdm 3165  ran crn 3166   o. ccom 3169  -->wf 3173  -onto->wfo 3175

This theorem is referenced by:  f1oco 3698  fodomfi 4546

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-fun 3187  df-fn 3188  df-f 3189  df-fo 3191

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