Theorem f1ococnv2 2817

Description: The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range.

Assertion

Ref	Expression
f1ococnv2	⊢ (F:A–1-1-onto→B → (F ∘ ◡F) = (I ↾ B))

Proof of Theorem f1ococnv2

Step	Hyp	Ref	Expression
1		f1of 2800	. . . 4 ⊢ (F:A–1-1-onto→B → F:A–→B)
2		ffun 2754	. . . 4 ⊢ (F:A–→B → Fun F)
3		df-fun 2432	. . . . 5 ⊢ (Fun F ↔ (Rel F ∧ (F ∘ ◡F) ⊆ I))
4	3	pm3.27bd 263	. . . 4 ⊢ (Fun F → (F ∘ ◡F) ⊆ I)
5	1, 2, 4	3syl 21	. . 3 ⊢ (F:A–1-1-onto→B → (F ∘ ◡F) ⊆ I)
6		iss 2599	. . 3 ⊢ ((F ∘ ◡F) ⊆ I ↔ (F ∘ ◡F) = (I ↾ dom (F ∘ ◡F)))
7	5, 6	sylib 173	. 2 ⊢ (F:A–1-1-onto→B → (F ∘ ◡F) = (I ↾ dom (F ∘ ◡F)))
8		fdm 2756	. . . . . . 7 ⊢ (F:A–→B → dom F = A)
9	1, 8	syl 12	. . . . . 6 ⊢ (F:A–1-1-onto→B → dom F = A)
10		f1ocnv 2811	. . . . . . 7 ⊢ (F:A–1-1-onto→B → ◡F:B–1-1-onto→A)
11		df-f1o 2437	. . . . . . . 8 ⊢ (◡F:B–1-1-onto→A ↔ (◡F:B–1-1→A ∧ ◡F:B–onto→A))
12	11	pm3.27bd 263	. . . . . . 7 ⊢ (◡F:B–1-1-onto→A → ◡F:B–onto→A)
13		forn 2789	. . . . . . 7 ⊢ (◡F:B–onto→A → ran ◡F = A)
14	10, 12, 13	3syl 21	. . . . . 6 ⊢ (F:A–1-1-onto→B → ran ◡F = A)
15	9, 14	eqtr4d 1131	. . . . 5 ⊢ (F:A–1-1-onto→B → dom F = ran ◡F)
16		dmcoeq 2573	. . . . 5 ⊢ (dom F = ran ◡F → dom (F ∘ ◡F) = dom ◡F)
17	15, 16	syl 12	. . . 4 ⊢ (F:A–1-1-onto→B → dom (F ∘ ◡F) = dom ◡F)
18		f1of 2800	. . . . 5 ⊢ (◡F:B–1-1-onto→A → ◡F:B–→A)
19		fdm 2756	. . . . 5 ⊢ (◡F:B–→A → dom ◡F = B)
20	10, 18, 19	3syl 21	. . . 4 ⊢ (F:A–1-1-onto→B → dom ◡F = B)
21	17, 20	eqtrd 1128	. . 3 ⊢ (F:A–1-1-onto→B → dom (F ∘ ◡F) = B)
22		reseq2 2576	. . 3 ⊢ (dom (F ∘ ◡F) = B → (I ↾ dom (F ∘ ◡F)) = (I ↾ B))
23	21, 22	syl 12	. 2 ⊢ (F:A–1-1-onto→B → (I ↾ dom (F ∘ ◡F)) = (I ↾ B))
24	7, 23	eqtrd 1128	1 ⊢ (F:A–1-1-onto→B → (F ∘ ◡F) = (I ↾ B))

Syntax hints:   → wi 2   = wceq 1091   ⊆ wss 1487  Icid 2057  ^◡ccnv 2409  dom cdm 2410  ran crn 2411   ↾ cres 2412   ∘ ccom 2414  Rel wrel 2415  Fun wfun 2416  –→wf 2418  –1-1→wf1 2419  –onto→wfo 2420  –1-1-onto→wf1o 2421

This theorem is referenced by:  f1ococnv1 2818  f1ocnvfv2 2920  mapenlem2 3385

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-res 2430  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

metamath.org