Theorem f1oco 2816

Description: Composition of one-to-one onto functions.

Assertion

Ref	Expression
f1oco	|- ((F:B-1-1-onto->C /\ G:A-1-1-onto->B) -> (F o. G):A-1-1-onto->C)

Proof of Theorem f1oco

Step	Hyp	Ref	Expression
1		an6 638	. . 3 |- (((F Fn B /\ Fun `'F /\ ran F = C) /\ (G Fn A /\ Fun `'G /\ ran G = B)) <-> ((F Fn B /\ G Fn A) /\ (Fun `'F /\ Fun `'G) /\ (ran F = C /\ ran G = B)))
2		funco 2696	. . . . . 6 |- ((Fun F /\ Fun G) -> Fun (F o. G))
3		funco 2696	. . . . . . . 8 |- ((Fun `'G /\ Fun `'F) -> Fun (`'G o. `'F))
4		cnvco 2520	. . . . . . . . 9 |- `'(F o. G) = (`'G o. `'F)
5		funeq 2683	. . . . . . . . 9 |- (`'(F o. G) = (`'G o. `'F) -> (Fun `'(F o. G) <-> Fun (`'G o. `'F)))
6	4, 5	ax-mp 6	. . . . . . . 8 |- (Fun `'(F o. G) <-> Fun (`'G o. `'F))
7	3, 6	sylibr 175	. . . . . . 7 |- ((Fun `'G /\ Fun `'F) -> Fun `'(F o. G))
8	7	ancoms 334	. . . . . 6 |- ((Fun `'F /\ Fun `'G) -> Fun `'(F o. G))
9	2, 8	anim12i 268	. . . . 5 |- (((Fun F /\ Fun G) /\ (Fun `'F /\ Fun `'G)) -> (Fun (F o. G) /\ Fun `'(F o. G)))
10		dmcoeq 2573	. . . . . . . . . . 11 |- (dom F = ran G -> dom (F o. G) = dom G)
11	10	cleq1d 1109	. . . . . . . . . 10 |- (dom F = ran G -> (dom (F o. G) = A <-> dom G = A))
12	11	biimpar 325	. . . . . . . . 9 |- ((dom F = ran G /\ dom G = A) -> dom (F o. G) = A)
13	12	adantrr 312	. . . . . . . 8 |- ((dom F = ran G /\ (dom G = A /\ ran F = C)) -> dom (F o. G) = A)
14		rncoeq 2574	. . . . . . . . . . 11 |- (dom F = ran G -> ran (F o. G) = ran F)
15	14	cleq1d 1109	. . . . . . . . . 10 |- (dom F = ran G -> (ran (F o. G) = C <-> ran F = C))
16	15	biimpar 325	. . . . . . . . 9 |- ((dom F = ran G /\ ran F = C) -> ran (F o. G) = C)
17	16	adantrl 311	. . . . . . . 8 |- ((dom F = ran G /\ (dom G = A /\ ran F = C)) -> ran (F o. G) = C)
18	13, 17	jca 236	. . . . . . 7 |- ((dom F = ran G /\ (dom G = A /\ ran F = C)) -> (dom (F o. G) = A /\ ran (F o. G) = C))
19		cleq2 1110	. . . . . . . . 9 |- (B = ran G -> (dom F = B <-> dom F = ran G))
20	19	cleqcoms 1104	. . . . . . . 8 |- (ran G = B -> (dom F = B <-> dom F = ran G))
21	20	biimpac 326	. . . . . . 7 |- ((dom F = B /\ ran G = B) -> dom F = ran G)
22	18, 21	sylan 343	. . . . . 6 |- (((dom F = B /\ ran G = B) /\ (dom G = A /\ ran F = C)) -> (dom (F o. G) = A /\ ran (F o. G) = C))
23	22	an42s 391	. . . . 5 |- (((dom F = B /\ dom G = A) /\ (ran F = C /\ ran G = B)) -> (dom (F o. G) = A /\ ran (F o. G) = C))
24	9, 23	anim12i 268	. . . 4 |- ((((Fun F /\ Fun G) /\ (Fun `'F /\ Fun `'G)) /\ ((dom F = B /\ dom G = A) /\ (ran F = C /\ ran G = B))) -> ((Fun (F o. G) /\ Fun `'(F o. G)) /\ (dom (F o. G) = A /\ ran (F o. G) = C)))
25		3anass 585	. . . . 5 |- (((F Fn B /\ G Fn A) /\ (Fun `'F /\ Fun `'G) /\ (ran F = C /\ ran G = B)) <-> ((F Fn B /\ G Fn A) /\ ((Fun `'F /\ Fun `'G) /\ (ran F = C /\ ran G = B))))
26		df-fn 2433	. . . . . . . 8 |- (F Fn B <-> (Fun F /\ dom F = B))
27		df-fn 2433	. . . . . . . 8 |- (G Fn A <-> (Fun G /\ dom G = A))
28	26, 27	anbi12i 369	. . . . . . 7 |- ((F Fn B /\ G Fn A) <-> ((Fun F /\ dom F = B) /\ (Fun G /\ dom G = A)))
29		an4 388	. . . . . . 7 |- (((Fun F /\ dom F = B) /\ (Fun G /\ dom G = A)) <-> ((Fun F /\ Fun G) /\ (dom F = B /\ dom G = A)))
30	28, 29	bitr 151	. . . . . 6 |- ((F Fn B /\ G Fn A) <-> ((Fun F /\ Fun G) /\ (dom F = B /\ dom G = A)))
31	30	anbi1i 368	. . . . 5 |- (((F Fn B /\ G Fn A) /\ ((Fun `'F /\ Fun `'G) /\ (ran F = C /\ ran G = B))) <-> (((Fun F /\ Fun G) /\ (dom F = B /\ dom G = A)) /\ ((Fun `'F /\ Fun `'G) /\ (ran F = C /\ ran G = B))))
32		an4 388	. . . . 5 |- ((((Fun F /\ Fun G) /\ (dom F = B /\ dom G = A)) /\ ((Fun `'F /\ Fun `'G) /\ (ran F = C /\ ran G = B))) <-> (((Fun F /\ Fun G) /\ (Fun `'F /\ Fun `'G)) /\ ((dom F = B /\ dom G = A) /\ (ran F = C /\ ran G = B))))
33	25, 31, 32	3bitr 155	. . . 4 |- (((F Fn B /\ G Fn A) /\ (Fun `'F /\ Fun `'G) /\ (ran F = C /\ ran G = B)) <-> (((Fun F /\ Fun G) /\ (Fun `'F /\ Fun `'G)) /\ ((dom F = B /\ dom G = A) /\ (ran F = C /\ ran G = B))))
34		3anass 585	. . . . 5 |- (((F o. G) Fn A /\ Fun `'(F o. G) /\ ran (F o. G) = C) <-> ((F o. G) Fn A /\ (Fun `'(F o. G) /\ ran (F o. G) = C)))
35		df-fn 2433	. . . . . 6 |- ((F o. G) Fn A <-> (Fun (F o. G) /\ dom (F o. G) = A))
36	35	anbi1i 368	. . . . 5 |- (((F o. G) Fn A /\ (Fun `'(F o. G) /\ ran (F o. G) = C)) <-> ((Fun (F o. G) /\ dom (F o. G) = A) /\ (Fun `'(F o. G) /\ ran (F o. G) = C)))
37		an4 388	. . . . 5 |- (((Fun (F o. G) /\ dom (F o. G) = A) /\ (Fun `'(F o. G) /\ ran (F o. G) = C)) <-> ((Fun (F o. G) /\ Fun `'(F o. G)) /\ (dom (F o. G) = A /\ ran (F o. G) = C)))
38	34, 36, 37	3bitr 155	. . . 4 |- (((F o. G) Fn A /\ Fun `'(F o. G) /\ ran (F o. G) = C) <-> ((Fun (F o. G) /\ Fun `'(F o. G)) /\ (dom (F o. G) = A /\ ran (F o. G) = C)))
39	24, 33, 38	3imtr4 192	. . 3 |- (((F Fn B /\ G Fn A) /\ (Fun `'F /\ Fun `'G) /\ (ran F = C /\ ran G = B)) -> ((F o. G) Fn A /\ Fun `'(F o. G) /\ ran (F o. G) = C))
40	1, 39	sylbi 174	. 2 |- (((F Fn B /\ Fun `'F /\ ran F = C) /\ (G Fn A /\ Fun `'G /\ ran G = B)) -> ((F o. G) Fn A /\ Fun `'(F o. G) /\ ran (F o. G) = C))
41		f1o2 2804	. . 3 |- (F:B-1-1-onto->C <-> (F Fn B /\ Fun `'F /\ ran F = C))
42		f1o2 2804	. . 3 |- (G:A-1-1-onto->B <-> (G Fn A /\ Fun `'G /\ ran G = B))
43	41, 42	anbi12i 369	. 2 |- ((F:B-1-1-onto->C /\ G:A-1-1-onto->B) <-> ((F Fn B /\ Fun `'F /\ ran F = C) /\ (G Fn A /\ Fun `'G /\ ran G = B)))
44		f1o2 2804	. 2 |- ((F o. G):A-1-1-onto->C <-> ((F o. G) Fn A /\ Fun `'(F o. G) /\ ran (F o. G) = C))
45	40, 43, 44	3imtr4 192	1 |- ((F:B-1-1-onto->C /\ G:A-1-1-onto->B) -> (F o. G):A-1-1-onto->C)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581   = wceq 1091  `'ccnv 2409  dom cdm 2410  ran crn 2411   o. ccom 2414  Fun wfun 2416   Fn wfn 2417  -1-1-onto->wf1o 2421

This theorem is referenced by:  isotr 2935  isotrALT 2936  ener 3313

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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