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 ∘ 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 ∘ G))
3		funco 2696	. . . . . . . 8 ⊢ ((Fun ◡G ∧ Fun ◡F) → Fun (◡G ∘ ◡F))
4		cnvco 2520	. . . . . . . . 9 ⊢ ◡(F ∘ G) = (◡G ∘ ◡F)
5		funeq 2683	. . . . . . . . 9 ⊢ (◡(F ∘ G) = (◡G ∘ ◡F) → (Fun ◡(F ∘ G) ↔ Fun (◡G ∘ ◡F)))
6	4, 5	ax-mp 6	. . . . . . . 8 ⊢ (Fun ◡(F ∘ G) ↔ Fun (◡G ∘ ◡F))
7	3, 6	sylibr 175	. . . . . . 7 ⊢ ((Fun ◡G ∧ Fun ◡F) → Fun ◡(F ∘ G))
8	7	ancoms 334	. . . . . 6 ⊢ ((Fun ◡F ∧ Fun ◡G) → Fun ◡(F ∘ G))
9	2, 8	anim12i 268	. . . . 5 ⊢ (((Fun F ∧ Fun G) ∧ (Fun ◡F ∧ Fun ◡G)) → (Fun (F ∘ G) ∧ Fun ◡(F ∘ G)))
10		dmcoeq 2573	. . . . . . . . . . 11 ⊢ (dom F = ran G → dom (F ∘ G) = dom G)
11	10	cleq1d 1109	. . . . . . . . . 10 ⊢ (dom F = ran G → (dom (F ∘ G) = A ↔ dom G = A))
12	11	biimpar 325	. . . . . . . . 9 ⊢ ((dom F = ran G ∧ dom G = A) → dom (F ∘ G) = A)
13	12	adantrr 312	. . . . . . . 8 ⊢ ((dom F = ran G ∧ (dom G = A ∧ ran F = C)) → dom (F ∘ G) = A)
14		rncoeq 2574	. . . . . . . . . . 11 ⊢ (dom F = ran G → ran (F ∘ G) = ran F)
15	14	cleq1d 1109	. . . . . . . . . 10 ⊢ (dom F = ran G → (ran (F ∘ G) = C ↔ ran F = C))
16	15	biimpar 325	. . . . . . . . 9 ⊢ ((dom F = ran G ∧ ran F = C) → ran (F ∘ G) = C)
17	16	adantrl 311	. . . . . . . 8 ⊢ ((dom F = ran G ∧ (dom G = A ∧ ran F = C)) → ran (F ∘ G) = C)
18	13, 17	jca 236	. . . . . . 7 ⊢ ((dom F = ran G ∧ (dom G = A ∧ ran F = C)) → (dom (F ∘ G) = A ∧ ran (F ∘ 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 ∘ G) = A ∧ ran (F ∘ G) = C))
23	22	an42s 391	. . . . 5 ⊢ (((dom F = B ∧ dom G = A) ∧ (ran F = C ∧ ran G = B)) → (dom (F ∘ G) = A ∧ ran (F ∘ 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 ∘ G) ∧ Fun ◡(F ∘ G)) ∧ (dom (F ∘ G) = A ∧ ran (F ∘ 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 ∘ G) Fn A ∧ Fun ◡(F ∘ G) ∧ ran (F ∘ G) = C) ↔ ((F ∘ G) Fn A ∧ (Fun ◡(F ∘ G) ∧ ran (F ∘ G) = C)))
35		df-fn 2433	. . . . . 6 ⊢ ((F ∘ G) Fn A ↔ (Fun (F ∘ G) ∧ dom (F ∘ G) = A))
36	35	anbi1i 368	. . . . 5 ⊢ (((F ∘ G) Fn A ∧ (Fun ◡(F ∘ G) ∧ ran (F ∘ G) = C)) ↔ ((Fun (F ∘ G) ∧ dom (F ∘ G) = A) ∧ (Fun ◡(F ∘ G) ∧ ran (F ∘ G) = C)))
37		an4 388	. . . . 5 ⊢ (((Fun (F ∘ G) ∧ dom (F ∘ G) = A) ∧ (Fun ◡(F ∘ G) ∧ ran (F ∘ G) = C)) ↔ ((Fun (F ∘ G) ∧ Fun ◡(F ∘ G)) ∧ (dom (F ∘ G) = A ∧ ran (F ∘ G) = C)))
38	34, 36, 37	3bitr 155	. . . 4 ⊢ (((F ∘ G) Fn A ∧ Fun ◡(F ∘ G) ∧ ran (F ∘ G) = C) ↔ ((Fun (F ∘ G) ∧ Fun ◡(F ∘ G)) ∧ (dom (F ∘ G) = A ∧ ran (F ∘ 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 ∘ G) Fn A ∧ Fun ◡(F ∘ G) ∧ ran (F ∘ 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 ∘ G) Fn A ∧ Fun ◡(F ∘ G) ∧ ran (F ∘ 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 ∘ G):A–1-1-onto→C ↔ ((F ∘ G) Fn A ∧ Fun ◡(F ∘ G) ∧ ran (F ∘ G) = C))
45	40, 43, 44	3imtr4 192	1 ⊢ ((F:B–1-1-onto→C ∧ G:A–1-1-onto→B) → (F ∘ 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   ∘ 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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